Intermediate Algebra Masterclass | Tahir Yaqub | Skillshare

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Intermediate Algebra Masterclass

teacher avatar Tahir Yaqub, I Teach Online

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Taught by industry leaders & working professionals
Topics include illustration, design, photography, and more

Watch this class and thousands more

Get unlimited access to every class
Taught by industry leaders & working professionals
Topics include illustration, design, photography, and more

Lessons in This Class

49 Lessons (6h 56m)
    • 1. Introduction To Algebra

      8:41
    • 2. ALG 11 LCM and LCD 1 C

      9:56
    • 3. ALG 12 Fractions C

      6:21
    • 4. ALG 13 LCM and LCD Method 3 C

      12:53
    • 5. ALG 14 Comparison Of Fractions C

      4:08
    • 6. ALG 15 Comparison Of Mixed Fractions C

      3:40
    • 7. ALG 16 Pre Requisite For Simplification Of Fractions C

      4:13
    • 8. ALG 18 Decimals To Fractions C

      15:53
    • 9. ALG 19 Checking Equivalent Fractions C

      2:29
    • 10. ALG 20 Addition And Subtraction Of Signed Numbers C

      12:22
    • 11. ALG 21 Multiplication And Division Of Signed Numbers C

      15:08
    • 12. ALG 22 Factorization Of Numbers C

      3:46
    • 13. ALG 23 Examples Of Prime Factors Part 1 C

      10:00
    • 14. ALG 24 Examples Of Prime Factors Part 2 C

      8:06
    • 15. ALG 25 GCF Part 1 Two Numbers C

      5:30
    • 16. ALG 26 GCF Part 2 Large Numbers C

      10:30
    • 17. ALG 27 GCF Of Algebraic Terms C

      2:41
    • 18. Irrational Numbers

      5:08
    • 19. ALG 07 Set Of Numbers And Rational Numbers C

      13:46
    • 20. Set of Real Numbers

      3:07
    • 21. Properties of Real Numbers

      11:28
    • 22. Making Your First Algebraic Expressions From Word Problems 1

      3:55
    • 23. Like Terms

      6:36
    • 24. ALG 06 Like And Unlike Terms Examples C

      13:33
    • 25. Degrees And Factors Of A Single Algebraic Term

      4:15
    • 26. Introduction to Exponents

      7:04
    • 27. ALG 30 Negative Exponents C

      5:45
    • 28. ALG 32 Zero Exponents And Exponents Of Negative Terms C

      7:34
    • 29. ALG 33 Perfect Square And Nth Roots C

      7:26
    • 30. Multiplication of Terms Having Exponents

      6:01
    • 31. Division Of Terms Having Exponents

      6:52
    • 32. Introduction To Equations And Their Solutions

      6:02
    • 33. Balance Method Of Solving Algebraic Equations Normalized

      20:38
    • 34. Reversing Method Of Solving Algebraic Equations

      13:44
    • 35. What Is Factorization And What Skills You Need

      5:05
    • 36. ALG 42 Terms Used In Polynomials C

      14:08
    • 37. Method of Factorization of Polynomials

      13:30
    • 38. Factorization Of Polynomials Using Identities

      7:27
    • 39. Synthetic Division:A Factorization method when one root is known.

      10:48
    • 40. ALG 35 Types Of Algebraic Equations C

      4:37
    • 41. ALG 37 Two Basic Methods Of Solving Equations C

      5:06
    • 42. 01 Method of Solving Equations with Grouping Symbols

      6:43
    • 43. ALG 36 Infinite And Unique Solutions C

      6:24
    • 44. 02 Examples of Equations with Grouping Symbols

      10:29
    • 45. 01 Introduction to the Language of Graph

      7:13
    • 46. 02 Derivation of Slope Intercept Form of the Equation of Line

      7:28
    • 47. 03 Properties of Straight Lines

      17:08
    • 48. 04 Parallel and Perpendicular Lines

      17:30
    • 49. 05 Solution of Pair of Simultaneous Equations in x and y by Using Graph of Lines

      3:06
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About This Class

Intermediate Algebra Masterclass:

This intermediate algebra course will help students  grade 8 to 10 to get a solid foundation in algebra and get ready for higher mathematics courses such as calculus.  

This Algebra course is designed and structured in a way which makes it easy for students to absorb the Algebraic concepts.  There are some inherent conceptual difficulties which high school students face when learning algebra. This course is designed carefully and keeping in mind all these difficulties. I tried to present critical algebraic concepts in a specific order. I am confident that this sequence will help students by making it easy for them to grasp algebraic concepts. These building blocks of algebra will not only enhance the skills of students but will motivate them to think algebraically. I have tutored high school mathematics for many years and have applied math in my engineering and research career. 

To learn algebra at their best, students should start thinking algebraically. I understand that if the concepts are presented in a scientific and organized manner with the precise and clear usage of the terminology, students start to get the bigger picture and this makes them think in algebra. They start to develop critical algebra understanding and even start to talk in x and y. This is the stage where learning algebra becomes their passion.  When they master the art of abstraction, they love it.  

For one who has absolutely zero knowledge of algebra, there is a pre-Algebra section in this course. The students who have a strong grip on fractions and number theory may skip this section.

For Home Schooling Parents

The course could be a valuable resource for people who are homeschooling their children and want help in algebra. The quizzes are interactive and parents can sit with their children when they are solving the quiz and can see the instant result at the end. They can then select course videos in the areas of weaknesses and ask children to watch that video.

Future Plans for the Curriculum:

My plan is to present the fullest possible curriculum for students of grade 8 to 10. The contents of this course will increase over time because I will be adding more video lectures and resources such as homework and quizzes. In the beginning, I will add one quiz for each section of the course but later my plan is to add more topic-based quizzes. The objectives of every section will be added clearly at the beginning of each section so that you know what skills you must learn during that section of this intermediate algebra course.

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Tahir Yaqub

I Teach Online

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Transcripts

1. Introduction To Algebra: hello and welcome. This post lecture is a brief introduction to algebra, and we will be learning some off the technology are terms used in algebra, and these are the words which you will be listening again and again in this course. So the first thing is a genetic expression. As you already know, The automatic expression is a statement in which we have numbers and operators. A few off the numbers and few of the operators, for example, five multiplied by two plus three minus four is an arithmetic expression. We have some numbers and we have some operators is a multiplication operator. In addition, Operator and the subjection or pretty and did our four numbers. So this is a statement which has numbers and operators. So if I replace any of these numbers, at least one of them with any letter, then this arithmetic expression will become an algebraic expression. For example, this fight I replaced this five with the letter X mike light by two. And remember, in algebra, we almost always do not put a multiplication sign. We don't put anything. Sometimes we can't put dot just to elaborate a little bit, but we don't put anything So X mark to play, but to be right, it as two x in algebra. So two x plus today minus four. This is no energy break expression. So we always put a number first and then the letter We have replaced one number with the letter, but this is still a combination off number and the letter toe extrusion Number X is a letter. First of all, what is this x this x in algebra not only X, but any letter. Ex wife. We use many letters in algebra X Y A B mostly values lower case letters, but you can use capital letter doesn't make any difference. So we used letters in algebra and this is the main difference between arithmetic and algebra. These letters stairs for numbers, as in this case, this extent for a number five. We know this number, but sometimes we don't. We don't know this number if we have ah, algebraic expression five X minus seven. I don't know what is X and again it find X only from an expression until this expression becomes an equation and how it becomes an equation. If I put an equal sign, any statement which has an equal sign becomes an equation. For example, if I write like this five x minus sale on equal zero, then this expression will become an equation. So equation has expression on both sides, and the left hand side must be equal to the right hand side. So what we are saying here is that if I multiply an unknown number X with five and subject seven from get product, I will get zero. So what is the value of X? We don't know yet, but we can find there are a fabric procedures and methods to solve these type of algebraic equations, and we will be doing this later in this course. But the purpose of this first lecture is to get your family with some of the words some of the terms, as in English terms, which we will be commonly using during this course. So know we talk about what is an algebraic ter toe. Algebraic term has three components. It has a letter. It is a number, a sign number which could be positive or negative, and it says an exponent. So in a get victor can have this front minus two experts to the power three. So in this these are a few off the components, often a Djibouti term. So what is this X? As I already mentioned, this X is an unknown number. But there is also named for this which is card base, that it has an explanation and he test Ah coefficient. So an algebraic term has a poem with the coefficient. He quite a present disco efficient with another with another letter. So I can say that e is a coefficient X is a wary about and and is an experiment. So I can say that in a generic term has a form off E X rays to department in we're is a number X is a number and could be a number. So when we write something like this, you say Al Jaber Tictail has a form e x ray to the power. And when we write anything in this form so what we're doing is we are trying toe generalized the things. So this is generality. We're trying to generalize the things and this is the way we sold problems in algebra. We try to generalize things and then try to achieve abstract solution. A more gentle solution to a problem. So this we will explain in that letter industry in this course. But at this time, these are a few of the things which I wanted to highlight in this for section of introduction. What is an energetic expression? What is an algebraic equation and what is in a diabetic? This is also a director, but the base here is X Y X Y is the base of the soldier Victor. Similarly, take another job, Victor, to be exodus to the power to Why is it so? This is also Najib Richter. So what about reading? These terms can be edit, are subjected. Our market plight are not like numbers. We can we can multiply any number. But what about these terms when we use these dumb in a statement are in an expression are in an equation. So I can me do operation on these times are not Can I add minus two x Q 25 x y What would be the result? Damn, it aired these three terms. Can I multiply these terms? Can I simplify these three times? So these are the different type off questions which we will be answering in our next reviews 2. ALG 11 LCM and LCD 1 C: So our next topic is least common multiple and least common denominator. So for example, we have two numbers, 34, and we have to find out what is the least number which is divisible by both 34 debt would be their least common multiple. So how to find it? Lets start with three. So find the multiples of three. So the multiples of three are, the first multiple is of course three, then six. If you multiply three by to get six, then nine, then 12, then 15. And what are the multiples of four? So of core four multiplied by one is 44, multiplied by two is 84, multiplied by three is 12, and then 16. So if we look at these two, the multiples of three and the multiples of four. The least number, which we'll find is 1212 is the least number, which is the multiple of both 34. So this is the least common multiple of 34. So what is least common denominator or least common denominator is same as least common multiple. But when we are talking about frictions and their denominators. So let us say that we have two fractions. One is two divided by three, and the other is three divided by four. So what is the least common denominator of these two fractions? So, least common denominator will be the least common multiple of the denominators. So least common denominator is simply the least common multiple of denominated. So this term is used only when we are talking about frictions. So these are two fractions. We have two by 33 by four. So the least common multiple of 34, which are denominators of these two fractions would be LCD. The only difference is that we use the term LCM when we are talking about certain numbers. We use the term LCD when we are talking about few frictions. So when the numbers are few, just two or three numbers, then this method is fine. You just find the multiple of the first number and the multiple of the second number. But if there are more than two numbers, are the numbers that large, then this method sometimes become very difficult, not difficult, but you can say the length method. So there is a simple method for finding the LCM, our LCD of numbers, and which we have used RAD in finding the prime factors. So this was, this was the method one. This is the first method of finding the LCME. The second method of finding LCM is this is a method to method two is you first find the prime factors are both numbers. Find the prime factors. So this is the step one. So lets find the prime factors of 34, the prime factors of 3r, three multiplied by one. This is the prime factor of three. And what are the prime factors are for? Prime factors of four are two multiplied by two multiplied by one. So these are the prime factors are for. So how do we find these numbers? So this method has already been described in another video. So you can watch that video to how to find the prime numbers. But basically what you do is tell you just put four here and find the minimum number which is divisible by four, which is two. Then two, then you find another number, minimum number to one. When you get one here, you multiply all these numbers and you get the setup prime numbers. So which is two multiplied by two multiplied by this one, which if you want to write, you can write otherwise 221. So after finding the prime numbers, what is the next step to find the LC? So this is the step one. Now step two is finding the maximum occurrence of a prime number in these vectors. So first we find out what prime numbers are occurring in the vectors of these two. So 321, so these are the three different prime numbers. So you can ignore one because one will not contribute towards LCM. One multiplied by guaranteeing is the same. So only 23, these are the two numbers, do prime numbers which are occurring in the factors of these two numbers. So what is the maximum occurrence of two? Maximum, what are the maximum number of times to occur? Some true occurred two times into factors of four. So the maximum occurrence of two is two. And what is the maximum occurrence of three? Maximum Auckland's after is just one. So now once you do, the LCM would be equal to you multiply to two times. Then you multiplied 31 times. So this is the maximum, maximum occurrence of two. So you multiply 22 times and then you multiply 31 time. So this is the LCM. So this is the second method of finding the LCM. You will find the prime factors first. Then you see the maximum occurrence of any number in those prime factors. So number to occur two times, number three occurred one time. So we multiply two times and 31 times and we get the LCM if the numbers are more than two, R if the numbers and large numbers. So then these two methods, sometimes there are more chances of error in these two methods, and these two methods sometime become very lengthy. So I'll give you an example. And we will learn the method to one more example, and then we will go and learn third method. So the second example of method to let us do it here. So we are trying to find out the LCM of 6927. So let us find the LCM of these three numbers. So first, find the step one is find the prime factor. So to find the prime factor of six, you can see the video. We are just repeating this again. So prime factor ofs seek the minimum number divisible by 60 is 23 times. Then the minimum number divisible by three is 31 time. So the prime factors of six are three multiplied by two, multiplied by y, and a few on. The prime factors of nine. Similar manner, 33 times. Then 31 day when we get one fee, God, the prime factors of nine, which are three multiplied by three multiplied by one. Similarly, the prime factors of 27. The minimum number divisible by 27 is three, because seven plus two, the divisibility rules. Don't forget divisibility rules because divisibility rules are the rules which we use to find the prime factors. So if two plus seven to plus seven is 99 is divisible by three, so 27 will be divisible by three. So nine times ten again three and we get three and then again three and we get one. So prime factors are 273 multiplied by three, multiplied by 31. So now the second step is we have to find out what numbers, what prime numbers are occurring in these three prime factors. So we can see that 23 is there and only 23. So how many times two, what is the maximum occurrence of two? So maximum what kinds of 2.0's In this six, which is only one time. What is the maximum occurrence off number three? So no maximum organism number three's three times in 27. So this is three times. So three is occurring three times. Two is occurring one time only. So our LCME will be p multiple a21 times, so only two and multiply 33 times two into three, into three, into three. So this is 631818 into three is twenty two fifty four. Fifty four will be the LCM. So in this example we have used method2 or finding the LCM, not this first method. Because in first method we find the multiples of all these numbers. And then we find that by visual inspection, what is the minimum number which occurs in the multiple of these two numbers? So these were the two methods and know we will learn a third method of finding the LCM, RLC D, which is the more simpler method, which is just like finding the prime factors and which you will feel very easy and comfortable. 3. ALG 12 Fractions C: Okay, let us talk about frictions are rational numbers. So friction is any number which can be represented in the form a divided by b. Friction is the, is any number which can be represented as a divided by b, where E is called the numerator. And B's card, the denominator of the fraction. And these numbers, for example, three divided by phi is a fraction where three is the numerator and five is the denominator. And any number which can be represented in this fraction form is a rational number. So I'm just writing rational numbers side-by-side so that you can just understand that all frictions are basically irrational numbers, not dealer fluid, medium important things about fractions. So the first thing you about friction, which I would like to highlight is card equilibrate and friction. I would like to explain this EQ here and friction with the help of this simple diagram. Let's say we have one rectangle and a directing. And let us say we highlight this rectangle into 1.5 of this rectangle is highlighted. So, so if we represent this first rectangle in the form of a friction, it would be equal to one divided by two. Because there are two equal parts. And we have taken the for a one part out of 2s, one out of two. Similarly, if you look at this rectangle and we highlight two of these parts out of four. So this is equal to two divided by four to parse out of four equal parts. But to assume that both rectangles are of the same size. So you can clearly see that these both represent the same, the half of the rectangle, but we're representing into different form. So it means that this fraction and this fraction, although they are represented differently, but they are similar. They represent exactly the same value. And this is true. These two fractions are equivalent. So how we get equivalent fractions? What are equivalent for action? So here is the answer. So if we have friction, let us say a divided by b. And we multiply and divide the friction. A multiplying and dividing it with one, we will get the same friction. Similarly, if we multiply and divide this fraction, we do. We still get the same friction. So it means if we multiply and divide a fraction by any number 1234 are any decimal number 1.41.21.3. If we multiply the numerator and denominator with the same number, we get another fraction which is equivalent to the phosphoric ship. Then we multiply these two with a, we will get two a divided by 2pi, two multiplied by a, two multiplied by B. This friction. And at this friction, these both are equivalent. Dare value is same for actions having same value. So if you look at these two. So basically what is happening here is if you multiply the numerator of war after foster friction one by two, you get two. And when you multiply the denominator with two to get four, you get this friction. So it means these two fractions are equivalent. So whenever you multiply a fraction, multiply and divide a fraction with the same number, you get the equivalent fraction. So y cubed infractions are important. Equivalent fractions are important because we sometimes have to simplify the frictions. For example, if you have a fraction like 24 divided by 36, this is a friction. And this is a very big enough. These are very big numbers. If we have to multiply this with numbers, let's say seven divided by four. So these numbers will be very big numbers. But if we know that this fraction can be simplified to a flexion with small numbers, then we can easily multiply it with salmon and for. So, how do we simplify the fraction? So before we look into simplification, we must understand another very important concept, which is how to factor a number. How to get the factors of a number. So when we are able to get the factors of a number, then we will be able to simplify the fractions. So this is what I am going to explain next. 4. ALG 13 LCM and LCD Method 3 C: So another example, find LCD of five divided by 73 by 59 by 15. So we will do this by the second method which we learned. So using method two, the fastest step is finding the prime factors of the prime factors of seven are just say E1, N1. The prime factors are five are 51, and the prime factors are 15, r, t, and phi. 1. Second step is what are the different prime numbers which occurred in these prime factors? So the prime numbers, you put prime numbers on one side. So prime numbers which occur in DC, that 357. And what is the maximum occurrence of these numbers? Maximum occurrence of these numbers. Any of these maximum occurrence in any of these. So three occurred only once, five occurred only once here, and once here, the maximum accuracy of five is also one. So in any one, we're looking at the maximum occurrence in any one. So seven also occurred maximum one time. So they all occurred one times, one times, one times N1 times. So therefore the LCD would be T multiply one time multiplied by five multiplied only ones, and seven multiplied once. So you multiply all these three and you will get there. Lcd, which is 105. This was another example though, the same method if we want to use on some large numbers, like for example, find LCM of 14202150. So now we have four numbers. So first, you find the prime factors of these numbers. So prime factors are 14. You can see that we do if you are not comfortable what I am doing. A video for prime factors. So the minimum number, which is divisible by 14 is 27, then the minimum number is say one. So when we get one, we multiply these two. So 14 can be written as two multiplied by seven. How you can put one as well. Then the prime factors are 21st, find the minimum number divisible by 22. Then again, minimum is 25 times. And know that minimum is five, which is one. So 20, the prime factors of 22 multiplied by two, multiplied by five. Similarly, prime factors of 21, minimum prime number is 37, then seven is one. Again, the prime factors are 50, the minimum number is 225, then five, because three is not divisible, five plus two is 77 is not divisible by three, so 25 is not divisible by three. So phi, the next is five, so five and then one. When we get one, we multiply these three. So the prime factors are 21, r three into seven, and prime factor of 52 into five, into five. To second step is we have to see what prime numbers, what different prime numbers that are occurring in their prime factors, and what is the maximum or currents. So we can see there two is occurring in few places, three is occurring in food places. In fact, only one place, And five is occurring in food places, and seven is also occurring in fewer places. So what is the maximum occurrences have to, in any of these, any of these, the maximum occurrence of two in any of these. So the maximum occurrence F2, can be found in 20. We're too occurred two times. Two times. What is the maximum occurrence of three in any of these maxima occurrence in any of these is only once one time. The maximum occurrence of five is two. Here. The five occurs two times in this one. So two times. And what is the maximum occurrence of seven? Well, once here, once here, so the maximum is still one, only one time. So now LCD of 14202150 equals to two times, multiply to 2x times, multiply 31 times. So multiplied 22 times, multiplied three times, then multiply 52 times multiplied by five, multiplied by five, then multiplies c11 times. So you multiply all these four into 312 into 5601625, three hundred, three hundred in to seven days, 2100. So this will be your answer. Use this method to find the LCD of reasonably large numbers. Like these were 414202150. But if the numbers are larger than these numbers, then this process could be a lengthy process. So for those type of situations, we have a tired method of doing this, which is basically a new form of the second method of finding the LCM, which we are going to learn next. Ok, let us. Do the same example which we have done with the method two, which is 14202150. And we have to find the LCM of these three numbers. So by doing this with method three, it will be easy for you to see the, see the advantage of this method. So this is our method three are finding LCM. So in this method, we are essentially finding the same thing, the maximum occurrence of any prime numbers. And we are using the similar type of formulation of the problem which we have done in prime factors. So what we're going to do is we're going to write all those four numbers here. Better if you write in ascending order, or desk Charles of a mistake. If you write in ascending or descending order by using the divisibility rules. And don't forget that divisibility rules are again applicable here. And we are using these divisibility rules to quickly find out which number is divisible. So what is the divisibility rule by too far too, if there is, at the end there is any even number r a 0. The number is divisible by two. This is not divisible by two. So this is divisible by two. So we take two here and seven times, ten times. Not divisible where two, so we'll keep it like that. And 25. Nor we still have ten, which is divisible by two. So we will be doing two until at least we have one number which is divisible by two. We will only use two. So we will take several like this, five. This was the number which is divisible by two. So no, at 21 is not divisible. We keep it like this. 25, since there is no 0 at the end are no even number. So no, none of these numbers are divisible by two now. So now we can move on to three. So seven is not divisible by 35 is not divisible by three. But 21 is because two plus one is three. The divisibility rule says that if the addition of the two integer is divisible by three, the number is divisible by three. So yes, 30. So we keep, say, 15 as it is, and then we find psi1 and 25 ICTs. Note no number is divisible by three, so we move on to five. So now seven, like this, five is one, then someone is like their five is 25 is five times. There is one number which is divisible by five. So we have 71, say one, N1, again, say 11111. So now we multiply all these numbers and we get the LCM. Very simple. So LCM equals two multiplied by two, multiplied by three, multiplied by five, multiplied by five, multiplied by 722 to four into 32 element 251625302, seven twenty one hundred. So essentially what we are trying to find here is we are dividing this until any one of these is divisible by two. So this is the maximum occurrence of two, which we were trying to find in the previous method. Then we keep dividing until anyone of this, any one of these, because we are taking any of these numbers. So the maximum occurrence of three is one. Then we find out that the maximum occurrence of five is two, because one number was there, 25, which was divisible by 52 times. So this is the maximum occurrence of five. And the seven. It has only one maximum occurrence one times. So this is essentially the same method, which is the method to, but we can quickly solve it because we are doing all these things simultaneously. So we are performing all these operations simultaneously. So this simultaneous operations give us an advantage in time. This is the best method of finding the LCM for large numbers. You must always know that divisibility rules for 235. And then the number will be reduced to very low numbers. And then you can immediately see if there is any 71113 or other prime numbers. Because you are doing this to repeatedly and due to this repeated division by two, the number will be reduced to much smaller numbers. And then you don't have to memorised divisibility rules for very large prime numbers. If you know the divisibility rules have to 35, you use this method and you will be fine. So these are the three methods of finding LCM. Our LCD, LCD is just when these numbers occur in the denominator of fractions. So we will use the same method. So I hope this makes sense and this is enough for LCM and LCD. 5. ALG 14 Comparison Of Fractions C: I want to highlight and other important concepts of comparison of frictions. So comparison of friction means to find out which fraction has the larger. And you then the EDR. This means finding which friction has the larger value are the smaller value. So for example, we are given with two eggs to frictions. One is two divided by Sarah, and the editor is three divided by 7. First, start with the simple example. So which friction is the larger fraction? But this is the easiest possible case when both have same denominator. We are comparing, we have to find out which fraction is the larger when we are comparing. The important thing is denominator. So denominator is same. Then the friction with the larger nominate numerator is larger. So it means this is the larger friction because the numerator is three. So if denominator is same, then larger numerator, the friction with large enumerator is larger. So this was very simple. When the denominator is same, the friction with the larger numerator has the larger value. No. Take another example where denominators are not same. Two divided by 55, divided by ten. So which one is larger? So if the denominators are not same, then we have to make the denominator seem so hard to do there. We have to use the concept of equivalent friction, which we studied earlier. Equivalent fractions. Because if we can make the denominator same, then we can use this above rule so that whichever has the larger numerator will be larger. So how to make that denominator c? So the easiest way is start with friction having the smaller denominator. If we multiply the numerator and denominator by the same number, by the same hole number. So we get the equivalent. No, we're not changing the value of this fraction because the value is still seem, we can cancel these two out. If we vision, we get to see and know we get four divided by ten. So this four divided by ten is same as two divided by five. But the advantages that no, it has denominator ten, which is the denominator of the other production which we are comparing to. And know what we have to do. We have to compare only five divided by 104 divided by ten. So our problem has been simplified that we have to compare these two using the first rule. Denominator is same. So that term with the larger numerator is larger. So this is our answer. 6. ALG 15 Comparison Of Mixed Fractions C: Okay, let us talk about comparison of frictions. Comparison of mixed fractions. So when we compare mixed fraction, let's say one friction is 123 and the other is 234. So in this situation, which friction is larger? Where there is a whole number. If there is a hole number, one has one and the other has to. So we don't have to do any further investigations because this fraction has a greater whole number. So this friction is definitely larger than the other friction. So it is very simple case. When one has a larger number and the other has a smaller hole number. The only situation where we will investigate this mixed friction is when the both have same whole number. For example, 12 by 313 by four. Nor we can not see as simply by looking at these friction which one is larger? So in this case, how we proceed. First, we have to convert both of these into improper fraction are common friction. So how we do that? Three multiplied by one, multiplied this, and then we add this numerator. We add this, and that will be your new numerator. So 313 plus two is five. So this is five divided by three. And the second is four multiplied by one is four, and plus three is seven. So seven divided by four. Now we have two fractions, and now we have to make the denominator seem weak means common denominator. Look and say. So how to make this denominator? Common denominator means same denominator. So we make them. A 13B explained this, that you multiply the first one, the denominator of the second, and also divide it so that you are not changing the value of the fraction. Similarly, WHO multiply the second fraction with the denominator of the first one, and you divide it with the same number so that you are not changing the value of tau friction because you can cancel the three out and still the friction is same. And you can cancel these four out and distill, the function will be the same. These two fractions are equivalent fractions. We can compare these two. So know what to do. Multiply the numerator together. So 2012, and this will become 221. And the denominator would be 12. So know when we have pool fraction with the same denominator. So which one is larger? The one with the larger numerator is larger. So once we have same denominator, then we can easily compare the friction. 7. ALG 16 Pre Requisite For Simplification Of Fractions C: Okay, now we discuss the simplification of fractions. So these are the things which we need to remember that these are the things which we need to solve our algebraic problems. So simplification of friction. It is one of the most important things you should be very comfortable with. So hard to simplify fractions, as I mentioned before you. Before you can do the simplification of frictions. You must understand how to factor large numbers. How to factor. This is the thing you should understand first. And before you can understand how to factor, you must know another very important thing, which is the divisibility rules. Large numbers. Because we're talking about simplifications. It means there are large numbers involved. And we have to somehow simplify those numbers. How to factor large numbers. And far, to simplify large numbers, we need divisibility rule. So this is how it works. First of all, you will understand what our divisibility rules. Then you will learn how to factor large numbers and find their smallest factors. And then you will be able to simplify fractions. So it is always very convenient and very useful when you understand the whole sequence of met. Technicalities are difficulties. Because once you know that what you have to know, what you have to learn, it will be easier for you to solve the problems. So you'll know divisibility rules. And I'm just going to explain what our divisibility rules. And then you understand how to factor large numbers. And finally, you will be able to simplify the frictions. So what our divisibility rules. So let's take an example. Suppose we have a number 24. So divisibility rules mean by looking at this number 24, you should be able to answer some of the questions. Is this number is divisible by two? Whether this number is divisible by 32, by divisible women's. If we divide this number by two, whether we will get of integer R naught, because we only are interested in integers. Only integers can simplify the light numbers. Only integers can simplify the fractions. We are not interested in decimal answers. So these are divisibility rules. By looking at the number, you should be able to understand whether this number is divisible by two, whether this number is divisible by three, whether this number is divisible by five, are whether it is divisible by seven or nine or ten. If you know any of these are few of these divisibility rules. Even if you understand some of these rules, you will be able to simplify this number. And this is a very small number by looking at the, you can know that, okay, this is divisible by two, this divisible by 34 and not five. But by looking at what about this number? For example, there is a number to 43. Whether this is divisible by two. Perhaps you no, no, no, it's not divisible because it doesn't have a even number at the end are 0. So these are the divisibility rules which we are talking about next, so that you can understand whether these 243 is divisible by three R naught, because after two, It's not divisible by two. Now we will look at three, then we will look at five. Then we can probably look at seven or nine. So these are some of the rules which we will be discussing now. 8. ALG 18 Decimals To Fractions C: So the third type of fraction is mixed fresh. Mix function is a friction where a whole number combined with the friction. For example, 11 by two. So in this case, as you can see, we have a whole number and a friction. So this is guard mix friction. In order to do any of the operation of this type of fraction, v must first can work. We must can work these two improper fraction. So how we do that? Very simple. We have to multiply the denominator by this number. And we add the numerator, and we get the new numerator. So for example, in this case, two multiplied by one is two and plus one is three. So this is our new numerator. Numerator, and the denominator stays the same. So this fraction equals three divided by two. So this is the way we can work this mix fiction into an improper fraction. And nobody can perform any operation on this friction. We can compare this with any other friction. Are we can find the inverse of this fraction or add or subtract, divide whatever. So let's take another example. Seven or five. So DC Comics friction. So this is equal to five multiplied by C1. And we add plus four. So 35 plus 40 is 39, so 39 divided by phi. So another example could be nine to three. So three into 927 plus two. So 29. But three squares. Very simple concept. Although you can convert this to decimal as well. Like 11 by two is basically equals one plus one divided by two. Which is one plus half, which is equal to 1.5, which is one plus half, and which is equal to one. Y and phi will get converted this to a decimal. And very important thing to note here is that decimals and fractions, they are, they can be transformed from one form to another form. You can convert a decimal number to a friendship and affection from a fraction to a decimal number two, you can convert this decimal to a fraction and a fraction to a decimal. And this is what we're gonna do next because we are trying to understand all the related concepts. And this is one of them that you can convert a fraction to a decimal, or a decimal to a fraction. And there are some common convergence, like 1 tenth are 100 beers, you should immediately realized it. What is the equivalent fraction of this decimal? So this is our next topic. So next topic is decimals and fractions. What is that relationship between decimals and fractions? So let us say you have a very simple fraction, one divided by n. So what is the decimal equivalent of this? So this is, as you know, I started from a very simple example because you are familiar with that. This is one end and 1 tenth is equal to 0.1. And the way we get it is you divide one. But then since one is smaller than ten, so we have to put a tortilla decimal and then we can put 0 here. And noise one. Therefore, this is equal to 0.1 or you can say 0.1. Similarly, one divided by Hunter is a fraction, which is equal to 0.01. And so you can find this is 100. One divided by 100 would be 0.001. With the same procedure we used in distributed in this TV. You can repeat this 401 thousand. So if you notice here that if you are dividing one and these are all Casey with numerator is one. So these are the simplest of all relationships between deflections where the decimal over the numerator is one, and so on. So you don't have to perform this. For these type of conversion. You can straight away write their Tokyo underwater, but ten is 0.11 to 100 is 0.01, and so on. Or can we go from there to there? And the way you proceed it, you count how many points decimal is away. After one. So this is one. So we will call this first 123. So theta, three decades afterwards there is a decimal. So you have to multiply this with 11000 because 100 has three zeros. And this way we will get rid of this decimal. So if we multiply this with 13 zeros, because for every ten multiplication does 0 goes here. If we multiply by ten, if you multiply by a 100, this will go this decimal peer code there. After this. And if you multiply by 1000 is decimal will come after one. So therefore, we are multiplying this by 100. But this multiplication must be Ben asked with their division of 100. So that we're not changing the value of friction because this one can be cancelled out. If we want. We can cancel these one hundred, ten hundred and that is our ten would be same. So no, this 11000 and multiply it by the numerator. Because if nothing is under the numerator, there is only one here. When you are multiplying two fractions, you have to multiply numerator with the numerator and denominator with their denominator. So because there is no denominator here, so you put one here. And now you multiply numerator with the numerator, you get one. As we planned, our plan was to move the decimal point and bringing it here. Therefore, we multiply it with, multiplied it with 10000. So we got one there. And on the denominator, one multiplied by 1000 days, one chosen. You'll see this either way. We got the answer. So this decimal is equal to this fracture. So this were the simplest case when. Numerator equals one. Now let us take another example. Convert two divided by five into decimal, decimal number. So numerator divided by the denominator. This is what fraction is. So since T2 is less than five, so we put a decimal here and then we put a 0. No file is four times five is 20. So we put forward the top and then we got nothing left. So the answer would be 0. Or another example. If you have zero-point 3-6, converted this to a fraction, or we have to convert this to a flexion number, fresh original number from dislocation to hear. So how many digits towards the right? So there are two digits. So two digits mean. We have to multiply it by a 100 because every ten multiplication will go one decimal, right? So what we do, we multiply 0.26 multiplied by a 100. And in order to balance this numerator, we also multiply this in the denominator. And the denominator of this term is one. So we are not checking any value. This is still my 3-6. You multiplied and divide with a 100 to get rid of this decimal. And at the numerator, we get P6, that this is what we were planning. And on the denominator, one multiplied by a 100 is 200. So this is the equivalent friction of this number. So we can actually simplify this fraction. And this is what we are going to do next. How to simplify this 36200 by a 100. So 36 divided by a 100. We find the prime factors of 36. So the same way, way. 36 divisible by two. Yes. Ut divisible by two? Yes. Nine, not divisible by three is three, and divisible by three is one. So we have to prime factor 36 equals one multiplied by two, multiplied by two, multiplied by three, and multiplied by 2D. And now if we have to find the prime factors of a 100, so starting from 250, zeros are divisible by 2285. No five plus 27 sword not divisible by three. So divisible by 55. And node five is prime over caught one. So this a 100 can be written as one multiplied by two, multiplied by two, multiplied by five, multiplied by five. So now what we have to do, we have to cancel out as many numbers as possible. So one is to 12 to two is to do nothing else. So now t multiplied by three is 95, multiplied by five is 25. So this is the simplified form of our statistic toward it by 100. Simplified form of this number. So it means zero-point 3-6 equals nine divided by 25. This is the equivalent fraction of this decimal. So in this exercise, we have learned two things. How to convert a decimal to a fraction. And then how to simplify that fraction. Using the prime factors to be able to get the question of simplifying of friction, where there are two large numbers, even larger than these two. So the best approach is to find the prime factors are numerator and the prime factors ever denominator. And then you can sell out as many common factors as possible. And then multiply the remaining of the factors in the numerator and the denominator and you get the answer. 9. ALG 19 Checking Equivalent Fractions C: And we're just wanted to highlight one more important thing that sometimes you have to check two flexion whether these two functions are equivalent or not. For example, today you divide it by 61 divided by two. Rather these frictions are equivalent or not. So this is the problem. So how to proceed with this? And there are two ways to do this. Two ways to check that whether these frictions are equivalent or not. So the first message method is, as I just explained, you simplify both of these fractions. If you get the same fraction, you simplify both of these fractions. And if you get the same friction, then both frictions are equivalent. For example, let us start from three to 30 by six. What can be divided by three? So 12. So the simplified form of this reduction is one divided by two, which is exactly same as this. So it means that both fractions are equivalent. So what is the second method? Second method is a simpler method. So what you do, you write both flexion side-by-side. And you cross multiply these two. The numerator of the first with the denominator of the second and the denominator of the first with the numerator of the second, cross-multiply. So you multiply three with two, you get six, and you multiply six with one, you get six. So if these two are equal, then the frictions are equivalent. So let's take another example. Two divided by 73, divided by five, whether these are equivalent or not. So you just, because they are already in simplified form, two is a prime number, seven is a prime number, three is a prime number and five is a prime number. So we cannot simplify it further. So they are already in simplified form. So how to check this? You just cross multiply. You multiply two with five and salmon with three. So two into five, you get 1070, you get 21. So these are not equal. So it means the frictions are not equivalent. So these are the two ways to check whether two fractions are equivalent, are not equivalent. So if you can simplify, you can simplify it quickly and see whether the answer is same. If we cannot simplify, just cross-multiply and see the numbers. 10. ALG 20 Addition And Subtraction Of Signed Numbers C: So in this lecture we are talking about signed numbers, signed numbers and numbers with signs. So every number in algebra has assigned. In this less than ten minute lecture, you will learn everything about signed numbers. And after this lecture there should not be any confusion. Award to handle how to handle signed numbers for algebraic operations. So why we are studying signed numbers? Because we want to add them. We want to subtract them. We want to multiply name, and we want to divide sign numbers. So these are the four operations. And as you can see, there is a number line on the screen. This number line is just for demonstration purpose, debt from on the number line, positive numbers are on the right side of 0 and negative numbers are on the left side of 0. And we are not using number lines to add numbers. In this lecture, we are studying how to add numbers without numbered lines, straight away, addition, subtraction, multiplication, and division of signed numbers. So today what we're going to learn is there are only three cases of signed numbers. There are only three cases. So the first case is when all numbers in any expression, in any algebraic expression you are trying to solve or simplify. All numbers are positive. So this is case one. And the second case is when all numbers are many div. And the third case is when few numbers of positive and negative. So my focuses very direct without any extra information, just we're focusing on how to handle these numbers. There are some numbers are positive, some are negative, all are negative, are all are positive. And the beauty of the algebra is that if you can handle positive or positive and all negative numbers, then you can handle when the numbers are positive and negative. Because this third case is essentially the mixture of first two cases. And I'm sure that you can handle all positive numbers. If I ask you to add three plus seven plus nine, you can quickly edit their dishes. 19, very easy. And let me tell you one thing. When all numbers m negative distribution is still the same. What example? All number than negative, minus three, minus salmon, and minus nine. What is dancer guesswork? Still 19, but with a negative sign. So the design team was on the right side of the number line somewhere here. But this 19 is on the left side of the number line somewhere here. So this minus 19 is much less number. And 19 is a greater number. But the process is really simple. If all numbers are negative, it looks like that you are doing a subtraction. But in algebra, this is the addition of negative numbers. This was the addition of positive numbers, and this is the addition of negative numbers. So these both are, can be considered as addition process. So addition of positive numbers. And what is this? This is also edition, but addition of negative numbers. So as you can see, these two cases are gone. You have already understood all these two cases. How to handle all positive and all negative numbers? No, the only case left is when few numbers that positive and flew numbers are negative. So how to add them? Let us see the same exemple. Normally make few numbers at one more number, minus four, plus it, minus three, plus, let say ten. So we have two numbers positive and two numbers negatives. So what you do in this situation. So the first step, although not necessary, but for simplification and for ease, you can do this. What do you do? First? Put all negative numbers together. Minus four, minus three, plus yt, plus ten. And know what you do. You add all negative numbers together, just like you did here in case two. And you add all positive numbers together, which is a piece of cake for you. Actually the both are same. So add four plus three is seven and put the negative sign here. And then ten plus 818 and put a positive sign here. Okay? Very simple. Now you can switch these position if you like, 18 minus seven. If not, you have to learn only this last step which is extra what to do here. So this is the only difference which you have to learn. And this you can understand on number line if you want. Otherwise you can simply, there is a simple method and simple procedure for this. So when you are, you have got these only two numbers. You have simply fight all the numbers and you ended up with these two numbers. So what you do, you always, always, always subtract. So only these two step, first step. Always subtract because if both numbers have different signs, then this is a subtraction procedure. You can see that you are adding minus 718. But because there are only two numbers and from class one, from year one, we know that 18 minus seven is a subtraction. Although you can still say it's addition of a negative and a positive number. But note let's treat it as a subscription for example. So the first step is always subtract from greater number. Are we subtract the smaller number? Does Somalia from greater? So in this situation, when you are left with only two numbers, one is positive and one is negative, you ignore the sign, always subtract some other number from the given number. So subtract seven from 18, you get 11. Bc. Do step one. And the second step is ported the sign-off. Poured the sign-off greater number. Besides this. So what is the nice sign of getting number positive? So positive 11 is your hands. Very simple. Addition and subtraction of signed numbers finishes at this point. This is the only thing you want to know. If all numbers are positive, add them together. Put positive sign. If all numbers are negative, add them together, put negative sign. If few numbers are positive, few numbers are negative. You first add all negative numbers together and put a negative sign. Add all positive numbers together. Put a positive sign. Then do this. Always subtract from greater to loyal number and put the sign of the greater number. That is it. This is the algebraic as addition and subtraction of signed numbers. So let us take some examples. Let's say we have minus 20 plus 13, minus 5x plus four minus one. These are flew signed numbers. So how to add them? This is case three, f2 positive, negative. So the fastest step not necessary, but you can do it. You put all negative together, minus 20, minus five, and put positive together. Put minus one here. Minus one is also negative. It doesn't matter in which order you put these numbers. So these are positive numbers. And these are all negative numbers. At all negative together, 20 ignored the sign, 20 plus five plus 126. No port the sign because all are negative, so the sum is also negative. No, add these together 70, put the sign plus which no, the two-step procedure always subtexts smaller from the larger, which is larger, ignore the sign. Ignore sign. Here. We will put the sign of the larger at the end. Ignored sign. Which one is the larger 26. Subtract from 2626 minus 179. So this is step one. Step two is for the sign of the larger number, minus nine. So minus nine is your answer. Okay? Very simple. No confusion. You just subtract because why we are subtracting? The question is why we are subtracting here? We are subtracting because one number is positive and one number is negative. One number is positive, and the other is negative. Because here we didn't subtract. All numbers are negative, so we add it. All numbers are portals positive. We added them together. But then we are left with only two numbers. One is positive and the other is negative, then the process is subtraction. Now, you can see that we are adding a positive number, a negative number. But in fact, what you are doing is a subtraction process. Because one number is negative and one number is positive. Sometimes you will get the larger number would be negative. Sometime the smell in number would be negative. In both cases, you will always subtract smaller number from the larger number and put the sign of the larger number. Very simple, no confusion. So I hope this makes sense. 11. ALG 21 Multiplication And Division Of Signed Numbers C: Now we discussed multiplication and division. Again, there are three cases. First days when all numbers are positive. Second is when all numbers are negative in number three tired days, when fewer positive and fewer negative. If all numbers are positive, whether you are multiplying them, are, you are dividing, they're, the result will be positive. So three multiplied by five is 153, multiplied by four, multiplied by five is 60. No matter how many numbers, if all are positive, the result will always be positive or negative. You might expect that if all numbers are negative, that is, there would still be negative. But no, it's not. And if all numbers are negative, the result could be positive. And is there could also be negative. Yes, if two numbers, there are, if you are multiplying two numbers, not all, but two numbers, not all, but both of them. We are talking about only two numbers. If two numbers are negative and they are being multiplied R divided, that result would be positive. 42 numbers. For example, if minus five is multiplied by minus one, that dessert would be plus phi. Both numbers are negative. But the result is plus phi nought. Let's talk about the third case first. Fewer positive and fewer negative. So again, we first talk about two numbers. Not few are two numbers. The simplest case, two numbers, one is positive and the other is negative. So when two numbers of different sign then multiply, that result is always negative. Multiply or divide. Or for example, minus five is multiplied by plus four. So this four is positive, five is negative, both have different sign. That is art is negative 20. So in second case, both have same sign, so that desert is positive. In the first case, the both have same sign. Both are positive, so that is there, it is positive. So we can simplify these two cases far to numbers only. For two numbers, not for many numbers if three numbers are being multiplied together, which we will discuss now. But before that we are talking about only two numbers. So we can see for two numbers. So there are only two cases. First case is when both have same size sign. Whether both positive, negative. And the second case is when both have different sign. So when we talk about two numbers, there are only two cases. So when both have same sign, that is act is positive. And when both have different sign, the result is negative. For addition, we treated many numbers like same or add it all together with positive, added all negative together. Very simple until we are left with one positive and one negative number. And then we subjected and put the sign of the larger number. Now here we are talking about two numbers. First, if porters have same sign that is positive, if both are of different sign, that is there, it is negative. And it does talk about more than two numbers. So I'll just talk about exponents. For example, I could divide minus two, raise to the power phi. And on my denominator, I have minus two raise to the power four. So exponent means that we are multiplying this minus 25 times. So now there are five numbers. So if we use our rules of do numbers, you have to do this many times because sometimes these x exponent can be very large numbers. So one way of doing this is this is not a good way, which we have learned that v is not good for exponents. And I'm gonna show you a very easy method to handle these exponent. But first, let's talk about what we have learned. If I put this minus 25 times and this minus 24 times. So one way of doing this is because these n minus two, we can cancel these out straight away. We can cancel these out straight away. Four, and only minus2 is left, so minus2 is dancer. But what, what about if it is minus three? Minus two raise to the power five. Divided by minus three raise to the power five. So in that case, how to handle this situation. So minus two, I just write down to demonstrate Q. First, then I will show you how to solve this very easily. And we have to write minus 35 times as well. So what we have learned is when we multiply two numbers, if the both have same sign that is written with the positive. So what we can do here is we can multiply these two together. And because both have same sign, deserting would be positive. Ten these two. And that is there will be positive. Then you will be left with minus two. And then you do the same in their denominator. Mind, sorry, 99 again. Nine from these, 29 from these two, and you are left with minus three. Now what we can see here is that you can handle these two numbers together. No. Because both have negative signs. And multiplication and division, the result would be positive. So I'm just writing here. So this, the result of these two will be positive two, positive three. So these negative signs, you can say that these negative signs in that denominator and numerator can cancel out. And then you will be left with 16 at the top, and 81 in the numerator and a 21 in the denominator. And you can multiply these two numbers together and you will get the answer. You see your answer. So one way of doing this is like this. No, I will show you a more direct method of doing this, which you might have already figured out. But I will just elaborate this a little bit further with a new example, but I'm just taking small numbers again. So minus two raise to the power five again. And this time three raised to the power three only. Not the best approach here is that since we have a negative number raised to the old power, this is an odd exponent. And we know that every two powers will make this negative positive. Because when we multiply negative two with the negative two, we consume two of the exponent's. Consume two of the exponent. So this will make it positive. So every two exponents will make it positive. So it means the four explanations will make this positive. But the fifth exponent will make this term negative. So if the power is odd, if a power of a negative sign is, or we can say that power of a negative sine is odd, then that sign of the desert in will be negative. Sign off the desert. And the sine is more important because we know the number, the certainties negative. And when the power of a negative sign is E1, this sign of the result is positive. So we handled the sign separately with the number. So we can write this question in this way. Minus means minus one raised to the power five. Multiplied by two raised to the power five. Feet two minus1 common from the, from the numerator. Actually we have just separated the sign from the number. So we have taken minus one separate and the number separate. And in the denominator we don't have to do anything. Theories to the power three is 27. Note we know that this is an odd power, so minus one raised to an odd power is negative, negative 11. We can omit. And two raise to the power five. 32 divided by 27. So answer would be minus 32 divided by 27. So this is the good way of handling. You should know that if the power is or of a negative number, then the sign of the desert and would be negative. And if the poverty is y1, the sign would be positive. So this is the correct way of handling exponents of negative numbers. So this is one more situation in multiplication when there are more than two numbers. Or duplication of signed numbers. When minus three is multiplied by minus seven and multiply it by eight divided by minus one, raised to the power three. So in this case, we can count some time if there is no exponent, but we can still count how many negative signs are in the top. So there are two negative signs. This is first one and this is second one. So it means we have even number of negative signs. So that exerted would be positive. How many negative signs that in the denominator minus one raised to the power three. So this is three negative signs. Three negative means odd. Odd power, or power means that dessert and would be negative. So we will get a positive result. And at the top, whatever is the number, we're not calculating this number, whatever. And negative lizard ten at the bottom. Of course this is one. So we can say that this is just one. So now we have two numbers. One is positive and one is a, one has a positive sign here that has a negative sign. So the rule number two and both have different sign that is written would be negative. And whatever is in the numerator will be your answer. 12. ALG 22 Factorization Of Numbers C: Okay, now we talk about factorization. So what is factorization? Factorization is on multiplication. So let us say I multiply two with 34. What I get is two into three into 42236 into four is 24. So this is multiplication. We are multiplying Fu numbers, two or more number numbers and we are getting an answer. So what if I was given this number? And I had to find these three numbers? Because these numbers are factors of this 24. So these are the numbers which are called factors of 24. These are the factors. Because if I multiply these numbers, I can get 24. Deal could be other factors as well. So factorization is on multiplication. We do something and we undo something. So we do multiplication. And then in factorization, we find the numbers, which if we multiply, we get a single number. So in the, in the questions are factorization. We will be given one number and we have to find few numbers, the factors of this number, it can resist that. Factorization is opposite of multiplication because when we say oppose, it normally means in wass. And inverse of multiplication is division. Did a few things to note here that if I multiply six into four, I will get 24. So it means six is also a factor of 24. Similarly, if I multiply eight into three, I get 24. So if I look at all the factors of 24, you will see that two is a factor of four. So the factors of 24 will be 23468. And even 12, if I multiply 12 by two, I get 24. So 12 is also effective. And of course, 24 itself is a factor of itself because printer for into one, we get 241 is also factor of any number. One can be affected of any number. So these are the factors of 24. So what it means is there could be more than one way to factorize and number. This is very important to note. Then there could be more than one ways. Factorize a number. To factorize a number. And which way is the best way I, which is the good way. 13. ALG 23 Examples Of Prime Factors Part 1 C: And Gil, non-linguistic few examples of a transition. It is factorize 363. So how to factorize this number? The method which is very easy to understand eighth, because we have learned that divisibility rules. So this method you will find very easy. So make two lines like this and write downs to 63 here. Now for start from two, whether this number is divisible by two and there is no 0 and no even number at the end. So the number is not divisible by two to know come to three. So the divisibility rule was if you add the number 30 plus six plus three. So three plus six plus three is 1212 is divisible by three, so the number is divisible by three. So report three and we divide this number by three. So 121. Now let's take about what is this 121. So remember, I told you about the perfect squares. The concept which we discussed earlier, the past perfect is choirs. So number 121 is a perfectly square. And this number is a perfect square of 11. Because you must, as soon as you look at this number, you should realize that this is the perfect square of 11. So it means if I divide this by 11, I will get 11. And that this 11 is a prime number, so it cannot be divided further. So except by itself. So we divide dilemma, the dilemma we get one. So when you get one here, it means the number has been factorized successfully. And no new marketplace. All these numbers. And these are the lowest multipoles West factors of these numbers and this number, so 53 can be written as one multiplied by 11, multiplied by 11, multiplied by three. And you can rearrange it in ascending order. One multiplied by three, multiplied by L1, multiplied by 11. So now we have used two concepts to factorize this number. One is the concept of perfect squares. You should know at least perfect requires a phosphor anti numbers. And the second concept we have used is the divisibility rules, which we use here. We use the divisibility rules. We use that this number is divisible by three. So by using these two rules, we were able to factorize this number and find its lowest factors. And now let's take another example, which is 540. So again, these two lines, two becomes 0. So 0 is divisible by two. So total 414 is 42, 7's are 40, and then 0. Now again is 0. So it means it is again divisible by Pu, one to be N1 non-zero tensor. 135 meu fi, noise not divisible by two anymore because no, at the end it is five. It is divisible by five. But first look at three, or 53819. So it is also divisible by three. So alluded divisible by five. But because it is divisible by three, If we add one plus three plus five, we get 99 is divisible by three, so it is divisible by three. So we will take the smallest number of 53, so we will first divide it by three. And now the fourth, 1121545, no, no, again, add 5499 is divisible by three, so the number is divisible by three. So we put three here. 15. Again it is six, so the number is divisible by three. So we again divided by three and we get five. And No, there is only one number which can do this, which is five. And we get one. Knowing multiply all these. And we get the smallest factor that 540. So we can write five for t equals to one multiplied by. You can arrange, you can skip the first, next step and you can arrange this. There are two 2's and 33, so I can write one multiplied by two, multiplied by two. Then three times 35. So these are the smallest factors of this number. You can see that if you, if you use the divisibility rule of 95 plus four, is 95 plus four is nine. And because it is divisible by nine, it means nine is also a factor of this number. But we are most interested in the smallest factors which we have obtained using this method. You want to give you a concept of prime factors. Yeah, factorize two numbers. One was 363 and the factors were 30 multiplied by L1 multiplied by 11. Also, you can put one. The second number was 540. And there were two rows and three 3's. And the five, these are the factors. The smallest factors. Now, if you look at these two factors, Nobel, notice that all these numbers are prime numbers. In both of these. These are all prime numbers. Prime numbers are the numbers which can only be awarded by itself. 123511. All of these are prime numbers, and therefore, these are doped Brian factors of these numbers. These are the prime factors of these numbers. And as I mentioned earlier, that these smallest possible factors are very important for us. Because these are the prime numbers they came out before the factorized. Here's the rule which you should always keep in mind when finding these prime factors, is that there is only one unique set of prime factors. So prime factors. But why'd you the unique solution of factorization of a number? 14. ALG 24 Examples Of Prime Factors Part 2 C: We have learned a few things and we want to make use of that. First, we have learned exponents. And all we have learned how to find prime factors of any number. So no. In this example, we have a number which is 19 to 0. And we have to find its prime factors. And then x plus two supplying factors in the form of exponent. So this is our task. So by using the same method, we can draw two lines on vertical and one horizontal. And porto a number here. And start using the rules of divisibility. So the first rule of divisibility we should try for two. So at the end, if it is 0 or an even number, the number is divisible by two. Soviet divide it by two to 980, then 122612. And density. Again, we divide it by two because that is 0 here. 2481616160, again record 0 so we can divide it again by two. So you see mostly after distributions the number becomes less than three digits. So you can, you don't need any rules for 78, these type of things. Normally, you only need to use the divisibility rules for 235 mostly. So now two, so 240. We can again luckily divide by two. So 1202 again, we get to six or 120, again divided by 2302 again. 15. No, we cannot do it, but pubic hair is 15 and we know the 2-tuple by three. So 355 is a prime number. So we can only divide it by five and we get one. Now, our solution is to start from here and go like this. And you can start from one. So start from the top. Port 1 first. So the prime factors are one, multiply it by two. Or many 2s, 1234567, 2's. So two raised to the power seven multiplied by three multiplied by five. So this is the set of prime factors for this number. Really easy. So once you apply the rules of divisibility, that number starts to go down immediately and it becomes easier for you to check for further 34 even if required other numbers. So this is the way to find the prime factors of any number. Let's take another example and find the prime numbers, prime factors of this number. 5670. The last number was quite easy when credit was 0 all the time. How this number cause? So first is to of course the current is 0 at time. So total 416 to 8162361025. So you've got to F35 noise no more divisible by two because it's not even number are 0 at time. So check for three. So what will check for three? We add the digits two plus eight plus three plus 510313518. So the number is divisible by three because 18 is divisible by 33680. So it means this number, this large number is divisible by three. So we can proceed with 339271334153515. Now again, we check for three. So nine plus four plus five is 18. Again, the number is still divisible by three. So 315. Again check for three. So three plus one plus five is nine, which is divisible by three. This number is divisible by 31. Then one cannot be two wires, so 0, then we borrow five, so 15. Now again, check for 31 as 0 plus five is six. Which is, again, you don't have to write all this. We can simply say five plus one is six. I'm just trying to show you so that you can memorize the procedure, but you don't have to write this. So six is the number will be divisible by three, or 33 is 91515, 3-5, 50. Norway or 35, we get eight. It means the number is divisible by three. But because five is a timed, so if five are 0 is at time the number is divisible by five. And we can see there it is seven. And seven is prime. Get one. So the prime factors of 56701212343 raise to the power four or three exponent or 105. And then Sarah, it easy. 15. ALG 25 GCF Part 1 Two Numbers C: So greatest common factor, also known as highest common factor, is the greatest number which is divisible by two or more numbers, which is a factor of two or more numbers. So highest common factor is same as GCF. So for example, we have two numbers. One is 181, is 144. So what is the greatest factor of these two numbers, which is divisible by both of them. So that will be the greatest common factor. So why we use greatest common factor and in what situations we need this, this concept can be used for numbers, as in this example, and can also be used for algebraic terms. When we're solving equations in algebra for any unknown term, x, r, any other variable. So during those solution procedures, during Dumas solution matters, sometimes we have perfect rise. And during the factorization process we have to take commons. And when we take common, we want to take the maximum number, which is common between two numbers, are between two algebraic terms. So these are the different situations where we find this concept helpful and this method helpful to quickly find out the greatest number which is common between two numbers are between two algebraic terms. So in this lecture we will talk about numbers, how to find the greatest common factor between two or more numbers. And then later in the lecture, we will discuss how to find the greatest common factor between two algebraic terms. So I am mentioning these, both these concepts side-by-side. So you can understand that the same concept of SCF can be used in two different scenarios coming towards the number, there are many methods to find the greatest common factors between numbers. So there are many methods. No matter is good or bad. Whatever you like, you can use any of these matters, whichever you find easiest. Who can use that method? And you will get the same desert. But there are situations when one method will be a little bit more easy to use. So I will try to highlight those things. So the first method is finding the GCF using the prime factors. And we have already discussed prime factors, how to find prime factors. And you can see it in any of the previous lectures. So for example, we have two numbers, one eighty and one hundred forty four. And we have to find the greatest common factor between these two numbers. So by using the prime factor, what we do is we find out all the prime factors are two numbers first. So you can use it separately and later I will explain how you can use this simultaneously. So find the prime factors of 180 divided by 2 first, we get 90, then still divisible by two. So we get 455 plus four is nine, which is divisible by three. So a number is divisible by 3155 plus 16 divisible by three. So the number is divisible by 35. And then we get one at a time. When we get one at nine, we're done. These are the factors or prime factors are 180. So we can write here 180 is equal to two multiplied by two, multiplied by three, multiplied by three. And, you know, we find the prime factors are 1424 in the similar manner. So to 18 to 93331. So we write down these vectors here 24 times, then 232 times three. This is 1 fourth, four. So not the greatest common factor. First, we have to find the common factors. So this two is common in both, these two is common in both. Know this tree. And these three, you can just mark it like this so you don't make any mistake. This three with this tree. So now we have four common factors. So our greatest common factor, because if these are the factors which are common, so the greatest common factor will be the multiple of these four numbers. So we multiply these two from this pair, two from the second pair, three for this pair, and three from disappear. 22 to four into 312 into 336 to 36 is the answer. So very easy. You just find the prime numbers of these two numbers are more numbers. And then you multiply all the common factors and get the greatest common factor. So this method, when there are only two numbers, you can do it separately when there are more than two numbers. I will explain next, Hartle D, more than two number using the same method. 16. ALG 26 GCF Part 2 Large Numbers C: So another example, or there are three numbers, 252396468. And we have to find the greatest common factor of these three numbers. So these had large numbers. You just put all these three numbers together. As we did in LCM. In LCME, we're looking for any number which is divisible by any of these. But now it's a little bit different. So you have to make a comparison between this and the LCME method because otherwise you don't want to confuse. So here what we do is we try to find out a number which is divisible by all of these numbers. So this is the difference. In LCM we were using any after a number which is divisible by any of these. But no, we are looking at a number which is divisible by all of them. So because they all have even number at the end to 68, so it means two is divisible by all of them. So we divide by two. If they're not, if there is no such number, we stop. We stop right there. So two so 1262 1s to 190 to 91816 to 182 to the n for no, Again, we have even number, then the number is divisible by two. So we do it again by 26399117. Now, these numbers are not divisible by two. Now, using the divisibility rule, we check for three. So six plus three is nine. So number is divisible by three because nine is divisible by three. So nine plus 91818 is divisible by three. So this is also divisible by three. So seven plus 18 plus 199 is divisible by three. So it means 117 will also be divisible by three. So we divide by 321333927. Again, we check and the numbers are divisible by three. So 71113. Now, these are prime numbers. So we have to stop here. You have to stop here because there is no such number, which is divisible by all of these. No such number which is divisible by all of these. So we stop here. And if we multiply these four numbers, which We will get the GCF. So GCF equals two multiplied by two multiplied by three multiplied by 32 to 4312 into 336. So this will be the onset. So the same method, but in a little bit different formation, you can use it for three numbers. Are more numbers. No, I will explain another situation where this method is not very useful. And you can realize it in a very first step of using this method. And then you can use another method, which is the third method. Second method, whatever you call it. Now, let us take another example. You have to find the GCF of these numbers, 8510 to 136153. So if we use the previous method, you can immediately see that the numbers, this number is not divisible by two. So we, we don't have to check all these numbers far too, because this is not divisible by two. So we don't have to check these for two. Node, check for 38 plus five is 1313 is not divisible by three. So a number is not divisible by three. So we don't have to check out that 4-3. Now go for 55 is at the end. So the number is divisible by five. But when we go on the second number, this is not divisible by five, so we don't have to check any of them. Number 45. Now what about seven? If we check for seven, the first number is not divisible by say one. So we don't have to check any other 47. So now it goes on and on. And the next prime number is 11. If you check for the level of this number is evenly divisible by 11, not divisible by lemma. So we don't have to check. So it means the previous method is not really useful in this situation. So what to do now? So now there is another method which we can use in this type of situation, which rarely occurs in algebra. But this is just a situation which can occur in arithmetic question are when you are doing some competitive exams, then you might find this situation because they select questions which are a little tricky. In those situations, you will find that debt method may not be very useful. So in these type of large numbers and kind of host dial numbers, which are not divisible by small prime numbers. If they are not divisible by small prime numbers, then we use another approach which I am explaining now. So what we do in this case is we divide the largest number, 153 in this case, with the second largest number, which is 136. It goes one time. 136. And we get 1717 is the remainder. Once we get the remainder, then we put 136 inside the second largest number and divide it by the remainder of the fastest step. And now it goes, it times 136. And nothing is the remainder. So it means that this 17 is the GCF, the greatest common factor between these two numbers. Between 153136, It is the GCF is 17 between these two numbers because 17 is divisible by both of these numbers. So no, we check this with the third number, which is 102. If 17 is divisible by 102, then this will be the greatest common factor between these three. And then we will check with 85. And if 17 is divisible by eight, then it will be the greatest common factor among all of these four numbers. So what we do next is we divide 102 with 17 and it goes six times. No, we divide. It means the 17 is the greatest common factor between these two and between these two. Nor we divide by 85 and check whether it is also affecting of 85. So now it goes 52, 53, and it goes 85, 86. So it means 17 is the number which is the effect of all of these activities, which is the greatest common factor. So greatest common factor equals 17. So this will be the answer. So this is a method where we do a step-by-step process. We just take the first two numbers. So if a greatest common factor exist, then we will find it. And if it doesn't exist, then these numbers are relatively prime. They are called relatively prime numbers means that there is no common factor except one. For an example, let's take 2845. So we take, because it's simple to take the first method or tool 14, then 27, and then seven is one, nor take 45. So five plus 49, so 3351. So if we write the prime factors of 28 is two into two, into seven, into 1453, into three, into five, into one. So you can see that it is the only common factor which is common between these two is 11 is really common to any two numbers. So these number, these type of numbers are called relatively prime. It is a term used for these number, these type of numbers. So they are called relatively prime numbers. So it means that there is no common factors except one, there is no other common factor between these two numbers. So using this method, we, you can, you can find the greatest common factors between lied numbers if one exists. 17. ALG 27 GCF Of Algebraic Terms C: Take an example, love. We have our term X4, y cubed, z is square. And another term, which is x five, y cube, said Q. And the third term is x q, y is it for. Let say, these are the three terms. And we have to find the greatest common factor. Because if he could take common, something common from these three, for example, these three terms are, let say adding together are subtracting, being subtracted together. So, and we have to find a common factor between these three terms. So what is the greatest common factor? What is the greatest letter and its exponent, which we can take common from the internal standards, you can only take common the minimum exponent present. So the minimum exponent you can take common, for example, x has an exponent of four in the first term, x has an exponent of five. In the second term, x has an exponent of three and the third term. So what is the minimum exponent of x? Three? So only three can be taken as a common. So we can pick X3 as a common. Similarly, what is the minimum exponent of y? So we'll treat all basis separately. So what is the minimum exponent of y? So y has an exponent of three in the first term. Y has an exponent of three in the second term, and y has an exponent of only one mean that third term. So only one can be taken as common. Similarly, if you look at zed, you will find that Sarah has a minimum exponent of two. So we can take zed two common because it has three here and four here. So this will be the greatest common factor. Greatest common factor of these three terms will be x cubed, y squared. So if there are the algebraic terms, you can take only the minimum exponent which is present in all of those terms. We'll do a lot of questions about these in quiz, where you will get a lot of practice. But the concept is very simple. When there are algebraic term, you can take only minimum exponent as a common. 18. Irrational Numbers: and know that we have defined rational numbers. We can see it there sort of regional numbers. Sort of rational numbers is represented with this symbol. Que and que ese in the form Here do very by B Such dead A and B belongs to belong to set off in two years so and B could be positive or negative vintages, but not zero. So we defined the set of racial numbers. Ask you cited Q is adorable. B and E and B both belong to set a vintages. No, everybody of number five bees, irrational numbers. So let's talk a boat. Ive additional numbers so irrational numbers are simply the numbers which cannot be represented in the form A divided by B. So all these numbers are irrational numbers. So we can simply say that all normal, rational numbers are original numbers and this is the way we defined irrational numbers. So there are few examples. So first example is the square root off. So this is equal to one point 414 to something, and this decimal never tell me names. So this decimal is No, don't me, Nick. Okay. And another example would be the base off the natural log. Ato e. He's 1/4 to 2.71 it Wait one it And it is also known terminating and already pretty. They didn't know the petition in more in this number and the third very famous example very well, nor number by by his by is a ratio ratio off a circumference diameter off any circle. So any circle if you take the diameter of the circle and divided with the s or you take the circumference of the circle and divided with the time heater value in the same units. So you get this number, which is and then never ending. So all these type of numbers which cannot be represented in the form is slash me. Our original numbers norm, additional numbers and real numbers can have only two forms additional numbers and no additional numbers. So we can we can easily defined the set of irrational numbers on we define it with a letter Capital P, and we say that be the set off all X such that X is so known fishnet, Do you remember? We have not yet defined real number, but we're very close. So know that we have said that original numbers are known distant numbers, no additional real numbers. So it means it should be clear to you. I know that real numbers can only have the additional and nor rest of numbers. So there is no other form off numbers or real numbers must be either original number. Oh, any original number divisional are no regional number, so real numbers are off only two types, original and irrational. 19. ALG 07 Set Of Numbers And Rational Numbers C: Okay, the sets of numbers. So the first set of numbers is the set of natural numbers. Neutral numbers are the numbers which we used for counting objects in everyday life, such as 123, all these positive integers. So all positive integers are natural numbers, 123 up to infinity. So how we represent the set of natural numbers? We represent with a capital letter N written in this manner. And normally start a curly bracket. Parenthesis, you can say. And then we say 123 and then we don't have to write all these. Just put dot, dot, dot, dot, and close the bracket. So this is the set of natural numbers. And you notice that 0 is not included in the set of natural numbers, but we do need 0. Because if we subtract two exactly same numbers, 1minus one, what we get, we get a 0. N is number. If we subtract the same number from itself, we kept 0. So if we add a 0 to the set of natural numbers, then we get the second set of numbers, which is the set of whole numbers. So whole numbers is a set of numbers which is nothing but a 0 included in the set of natural numbers. So this is represented by w. And start the curly bracket and start from 012 and then put dot dot. And this is the set of whole numbers. Very simple. So set of natural numbers and set of whole numbers are very simple to understand. There is no confusion at all. Third one is the set of integers. And to understand the set of integers, district called the number line must have seen this number line in year six or so. And the center of this number line is the number 0. And on the right side are positive integers, 12. 2d. And on the negative side, the negative integers now minus one, minus two, minus three. So from 0. But this direction is ever set of whole numbers. But we haven't talk about this side of the number line. These are negative integers. And we combine all this. When we combine all these, the whole numbers means the positive integers and the 0. These are the positive integers. Positive integers, then Zillow and the negative integers. So when we combine all these numbers, we get the set of integers. So a set of integers is represented by set. And this includes phosphor dot, dot, dot for the negative infinite. Then feel negative numbers. Then 0, then feel positive numbers. And then look and put this daughter gain. So this is the set of integers. Really simple. You just need to add negative integers to the set of whole numbers and look at the set of integers. Nor the fault type of numbers are rational numbers. And just look at the word ratio. Additional numbers are those numbers which can be represented in the form of a divided by b with two conditions. Number one, that ANP should be integers. Positive or negative, doesn't matter. And second condition is that b should not be equal to 0. Because if B is 0, then the value is indeterminant. We cannot find the value of Edouard by b. So all those numbers which can be represented in the form of a ratio.edu or a by B are rational numbers. So some numbers are quite clear. For example, two divided by 450, divided by ten. For you, all these numbers, they are clearly rational numbers. But we're double sum of decimal numbers, such as the zero-point here to 1.5. So what about these types of numbers? Are the rational numbers are not rational numbers. And also there are some other numbers. These are the numbers we, first, these are the numbers which it should determinates. These that terminating decimals, because after two decimal there are no more decimals. Similarly, this is a terminating decimal number. There would be more decimals. But if the decimals are terminating decimals, then visit another story. But there are some numbers which never terminate, such as 0.333 and it goes on and on. So these are non-terminating numbers. And if you look carefully that this tree is repeating, So therefore we will say that this is a non-terminating because it never terminates. And known DPT has already repeating. These are non-terminating repeating decimals. And you can have some more examples of this, such as if you look at the number one divided by seven. This is a rational number of course. And this will be 0.1457. And it repeats again, 14 to wait 57. So this is also an and it never terminate, it goes forever. So this is also a non-terminating and repeating decimal number. But this I have given you an example that I can see that because this is a fraction, so this can be, and he presented as a rational number. This is a rational number. So it turns out that all known terminating repeating decimals are also rational numbers. All those decimals which are non-terminating but repeating, there is a repetition. And these repeating numbers are written normally in this form, 0.3 with a bar on top. It means the three repeats forever. Similarly, this number can be written as 0.14257 with a bar, because all these 60 years to repeat forever. So it turns out that all non-terminating but repeating, non-terminating. But repeating numbers are rational numbers. And these numbers are clearly rational numbers. What about these numbers? These are terminating number, terminating decimals. These are terminating decimal numbers. Still look or 0.8 to 0.82 can be written as 0.82 multiplied by a 100 and divided by a 100, which is equal to 200. So this can be written as a fraction. So it means or terminating decimal numbers can be written. So it means all terminating decimal numbers can be written as a fraction n. These are all rational numbers. 1.25. Again, this number can also be written in the form of friction, which would be equal to after simplification 125 divided by 100. It means all terminating numbers are rational numbers. Terminating decimal numbers are rational numbers. How long it may be. For example, if it's a very long number, let us say is CDO wine 1256. It's not repeating. But after a certain time terminates, They didn't know dot-dot. It doesn't go further. Ten minutes here. Ten minutes here. So anytime we knitting number, we'll be a rational number. Any decimal number which terminators is a rational number? Similarly, any non-terminating but repeating number would be a rational number. And these are very simple that they are already in the form of a friction. And b is not equal to 0 and both a and b are integers. So these are clearly rational numbers. So there are these three types of rational number. These three types of numbers are rational numbers. 20. Set of Real Numbers: So once we have defined relational numbers and irrational numbers, these two type of numbers basically cover or type of decimals where the terminating, non terminating, repeating our nor deputy. So it means all type of decimals that cover and therefore, if we add just I mean, if we aired the sets off real number and the set off, if we had the set off additional numbers in the set of irrational numbers, we will get the set of real numbers. So basically real numbers are or decimal numbers. So all decimal numbers are basically real numbers. So we define the set of real numbers. The Capital letter are written like this. RTZ Corp set of all Excited X has a decimal representation. It's one thing you should notice that all in cages and zero have decimal representation. One came bitterness 1.0, and you can't get Internet 0.0. So the definition of real number sets it is the same that feel numbers are all decimal numbers. So this concludes our discussion about itself, the real numbers and no, I was sure your diagram, which will explain this to you in a visual manner. So this diagram shows you hold. These sets are related to other sets. So as you can see that natural number, the set of natural number, he's the smallest, say said, And all nature numbers are essentially whole numbers, and similarly, all whole numbers are integers, and all in teachers are additional numbers. But there are some other types of numbers which are original and all national and plus irrational numbers didn't make the set of real numbers, so it is very self explaining to diagram. It's very simple diagrams and there are other in other types are type of number, which are complex numbers where those numbers are. We're not discussing in this court because this is an advanced topic and this is out of scope of our course. 21. Properties of Real Numbers: in this lesson, we're going to be looking at some off the basic properties off real numbers. So these properties are total 11 properties to closure properties. One is for addition and one is for multiplication. Similarly to commit rated properties one for addition, one form or application. And we have five of these one for addition, one for multiplication. Then we have a distribute to property. So what are these properties? So let us assume that A and B are to real numbers belonged to Shut off your numbers. So the closure property Saiz that if we add these two numbers and let us say we get another number, sees the addition of the some of these two number so close your property says that if he and be a real numbers, then there some will also be area number. Similarly, if we multiply A and B and NB aerial numbers and for example, their product is any number D, then the will also be aerial number. So this is a close your property of for additional disclosure property of March repetition . Then comes the community property. So let's take an example that if you add two plus three you were reading three into two are you were three plus two. You were too into a tree. We didn't make any difference. No. So this is what the community property Saiz that order off addition is not important. So if there are two numbers and be so a plus, B will always be a Quito be blessed. A. You see the community property of addition similarly, for multiplication. If a few more to play do with three are three with two because in algebra be used door for more dedication, so that is that it will be six. So the community property of multiplication says that order is not important in Toby will always be equal to being it. But this is only for addition and multiplication. There is no such thing for our subjection. A minus bi is not equal to B minus e seemingly here to order by B. It's not equal. Toby do anywhere. You until, of course, is one. And B is also one. But we're not talking. We are saying that A and B are two different numbers, so even Toby is equal to be into it. So this is where the communicative property says What is associative property? So the associative property Saiz that you can group the numbers in addition are in multiplication. However you like so there to say we have three numbers. So associative property says that you can ed e N b and then you can add c are you can group B and C and then you add a So this is group because you know that we perform the group operation first. So associative property say that you can group the number. In addition, however you like you can group you're reading three or more number. You can put the brackets anywhere you want. You can make any group within the addition. Similarly, if you murder plain A, B and C so you can put bracket here are you can put bracket on B and C it will not make any difference. A begin to see will be equal to and Toby C So this was associative property. So what is identity property? Identity property is basically saying that if there is a number A and we airs, you don't do any of this number, so the number will not change. Number will stay the same. So This is the identity property that zero are due to. Any number does not change the number. Similarly, one multiplied plenty number will not change the number. If you multiply, one with the number will stay the same. Which is it So a plus zero will be a and a multiplied by one will be it. So if we are zero to a number, the number will not change. If you multiply one toe a number, the number will not change. This is identity Property number five is the inverse property for addition and multiplication. So this says that if there is a number a then there exist a number minus e so that if we add these two number, we get zero so he bless minus e will be a zero. So this week would be positive Could be negative. This minus there could be positive could be negative. And if you don't understand this, you can just go back to your number line On this number line, you have zero in the middle. Want to try? Just put three numbers only. So if you dig a was one, then minus a, we'll be minus one. One is the a then minus a minus one. But if you take equals minus three, then minus a will be three. Got this number? There has a other number which is additive in words. These two numbers are made it even worse off each other. So for any number A, there is a number minus a so that if we add these two number that is there to zero. So adding e and minus is seem as subjecting a from a. So if it is written like this, we will say we're subjecting a from A If we didn't like this, we're adding here to minus E. In both cases, there is there will be zero because if you aired minus tree to tree, their third will be zero. But if the subject three from tree there is ritual with zero. So this is in west property of addition similarly for multiplication. If there is a number A, then there exist a number one divided by a the reciprocal off a also card mark implicated in worse. So if there is a number a, then they'll it will be a number one. Do. Everybody is such there. If we multiply these two numbers a more deployed by wonder where the way we there's a reciprocal the result will be one. So, for example, is it so if the reciprocal will be wonder where to buy it So if we multiply eight with one door debated one But here there's that will be one phone number is two So there will be a number Wonder what about a Which is wonder what about two. So I just want to show this number so that you can easily figure out. So if you're is to for example, this is your a Then wonder where everybody is half which is somewhere here So this is your a by two If it is to wonder where the way is half what it will be here somewhere smart implicate different wars So Marty, pretty creative in worst an idiot Even worse they're covered in this in worst property said that if you add edit even wants to a number a exactly zero If you multiply your number with their smart, typical different word there is there will be one. So the last property is distributed property so technically in this thing So they're distributive property supposed there are three numbers A, B and C and they're written like this is being multiplied by the US addition off BNC is multiplied by B plus c. So this is a group. This is a bracket. So the stability property says that you can distribute this number or these two numbers. So you first multiply every B, then it egg with the product off E N C. Then you might play It would see and you had these two. So this is also called opening the Records. So this property is basically telling us how to open the brackets like associate. The property was telling us how to move the record. You can move the bracket anywhere you want. If you are reading, are multiplying ABC So you can write like this are ABC You can write like this boat will be equal. Similarly, the distributive property telling us how to open the records. So if a multiplier will be policy So this is a very open the bracket. Distribute this a over B and C So am I prepared baby plus a murder probably see. So they are torture, 11 properties off numbers. So you better memorize this thing because you will be using these properties throughout this algebra course, he said. The basis off number, operation, any operation or numbers, any operations on variables. We will be using this again and again. We will be opening breakers. We will be taking common so all these type of things it will be doing. 22. Making Your First Algebraic Expressions From Word Problems 1: so No, These are the two examples of how to make algebraic expressions. So the first is Adam had 23 Candies. His friend gave him some more se x right. An expression for the total number of Candies Es So he had 23 in the beginning. Plus, his friend gave him X Candies. So this is a statement for total number of Candies. This is a general statement. Whatever his friend gave him, we don't know, but we can put here and we could get dancer go. The number of Candies. Yes. So this is a general expression. So this is what we do in algebra. We make some general expressions, and then we sold for their Let's take another example, I don't work. It's our spot. We call many hours. He will be working in four weeks. So if he works X hours in one week, So he will be working for murder right by X hours in four weeks time, and we don't put more duplications. And it's simply for our forex so emitting a scenario that he can change his working hours after every four weeks. And you are suppose making his paycheck every four week and you are trying to calculate the number of hours, so a four week is our target time. So let us imagine that the 1st 4 weeks he decides to work 20 hours per be so the total number of hours it would be foreign troop 20 which is 80 hours. Next four weeks, he decides to work 30 hours. We still have the same statement for Index because we're calculating his total number of hours for over imaginary scenario off, Let's say, giving him a pitcher every four weeks. So four exodus statement. So if we decide to work 30 hours the total number of hours after four, it would be 14 to 30 which is 1 20 So you know, you can get an idea that whole algebra can generalize the things I can give you a general expression, which you can use repeatedly for the same situation with a different variable. So X is a variable, the number off hours. He can decide whether he's working 20 hours, 30 hours, but he can decide only after four weeks. So in this situation, four X is a general statement for total number off hours, so two more examples. He bought something for X dollars, but she sold in $339 homered property she made from the stick. So the prophet is simply the seal prize money The purchase price. So 3 29 is the same price minus purchase price. Waas X So this is the general straight man for profit. So whatever is the purchase price, this will be the answer for profit. The last example. Our team consists of P players, the team with the cash spread of $750. This prize money will be distributed equally among all the players. How much money is play every day, The amount each player will get equals 752 or two But he this is the amount each player will get. So these were some of the examples of whole tomato algebraic expressions. 23. Like Terms : Welcome back. We're still in introduction to give her a class. And the next thing is like terms in a Gabriel. You must understand what are the like terms. So what? They're like terms. So first letters every ways again. The term fabric algebraic term has ah far e exodus to the power and we're X could be abyss . So what are like them like? Terms are those are separate terms which have seen base and same explanation. For example, one term we have three x rays to de power to me. And there is another term which which is minus nine x rays to the power three. So both of these terms have same based, which is X have seen exponents, which is three. So therefore, these two terms are like terms, so they're different only in coefficient. So I like terms differ only in coefficient. The dumps can have more than one letter as a base. For example, e minus fight X Y squared and Google ex away squid both have same base, which is X Y, and have same exponents which is to wanted The white part of the base has their exponents. If we compared these two terms. Let us compare this with another term which is minus seven XY squared y No, the basis saying But exponent is not because in this case, X has their exponent of two while in this case what he has an exponent off. So not only the base shoots should be the same. But also the exponents should be the same. Now let's take another example then whyy squared X If you look at this no, the basis Boy X what has an exponent of to the next has an exponent of one all these three terms this one this one and this one All these three terms are like terms. The order doesn't matter whether you have expressed and then y squared are you have y square first and then next doesn't matter. So the concept of like terms is very important. You should be very comfortable with exponents and I have a few lectures about exponents later in this course. So the exponents off every letter should be the same. But for example, if I tell me is one divided by X is quiet and other enemies minus five. Exodus to the power minus two. These two terms look different. But in reality, in both of these, the base can be written in a similar form. This is one Doherty with excess square X rays to the power minus two. And no, if you look at these two terms minus five X rays to the power minus two Elected to the power minus two. You can easily recognized that both are like terms. Both have seen base. I m seem exponent, only different coefficient. The coefficient of this is one. While the co option of that there is minus fight. As an algebra student, you must understand that these are just two different ways of writing the same base. So we will talk about these explanations. And what are the various forms off exponent later in this course? No. The next thing is why were interested in like terms we're interested in like it because only, like terms can be added are subjected from each other. So if we have minus five x Esquire, why plus three X is quiet. Why? So we can add these two terms. Forget it. So because they have the same based on exponents part the only different in coefficient. So we just add minus five and three, which is minus two, and I will also explain this addition and subtraction off the sign numbers later in this course. But for the moment, you can simply say that subject for a three from five and put the sign of the bigger numbers so minus two Exit Square. Why would be the answer? So only like terms can be added are subjected. We cannot enter subject unlike terms. So, for example, we have two terms minus X is quiet plus X cubed. So these both of these terms have seen based but export. It is different. So these terms, although being added but they will stay like this, they cannot be further simply fight. They will stay like this. So when we say that we can add are subject only lifetime, it means that we can actually simply firelight terms to get a single edge every term, while we cannot simplify the unlike terms and they stay the same. So this is the reason why we were interested in like terms and know you are able to recognize the like terms in any algebraic expression 24. ALG 06 Like And Unlike Terms Examples C: Ok, we have learned the concept of like terms. So let us do few examples. Because this is the concept you should be very clear on. Students. Make lots of mistakes by unable to recognize the liked and performing some operations which are not allowed. Six x multiplied by three in multiplication and division, there is no problem. We can multiply or divide unlike terms. So 663, they are both unlike terms because it has a base of x, while this has no base, it is a constant. Three is the constant term. These are unlike terms, but we can multiply them. There is no problem. Six Martin number will be multiplied by numbers. So six multiplied by three is 18. And x is the same. 6x plus three. Again the same two, unlike terms, but now the operation is addition. And for addition, we cannot add two unlike terms. So what it means is when we say we cannot add, it means we will keep these two terms in the same form as it is. So we are not removing them, we cannot add them. But we will keep those two terms as it is to the next step. If there is a question, we are solving a question. And there is a situation in which two, unlike terms are being added together. So we will keep those terms at the same stage. We will not simplify them. So this is what it means. So the answer would be six x plus three. Keep them same because they are unlike most six x plus 3x. Now look at the base. The base is same. The exponent is one because if no exponential, which is one, so base and exponent is same. So these two are like terms. So for like terms, we only add the coefficient. So six plus three is nine. So nine and the bass stays the same. So nine x would be the answer. No, 6X minus six Y. So these are clearly unlike terms. Both terms stay the same. So six x minus six y 2x y multiplied by two in multiplication, no problem. So two x, y multiplied by two. So only integers multiply when there is no base. So there is nothing besides these two. So only this integer will be multiplied by this term, this whole term only the integer, because it has no variable term. So 2S multiplied to X, Y multiplied by two is four x y, no, 2x y multiplied by 2x. No, we have one variable in the second term. These two are still unlike terms. But because we are doing multiplication, so no problem. So integer multiply it by integers. So for X multiplied by X, when there is x in both terms, we have to multiply both x together. X multiplied by x is x square or x raised to the power two x. If x is multiply two times the exponent of x will be true. If x multiplied three times the exponent x will be three, and so on. So for x is squared and we put y beside this. Now the next 2x y multiplied by 2y. Unlike terms, again, only one letter here. There are two letters here, so they are unlike terms. To multiply it by two is four x, there is no x on the second term, so we keep only one X from the first date is one while the second term and one way in the first term when multiplied together. So y will be multiplied by y. So y raised to the power two. Now 2XY plus 3Y X. So are there, like, you can pause the video and think, yes, these two are like terms because the order does not matter. And we will soon see this when we will study the number properties. And because it's an addition operation, so two plus three is five. So five x y is a correct answer. And also five YX is also a correct answer. Because the order of these two does not matter. The last example, 2x multiplied by two, y, 22 multiplied by two is four. And X multiplied by Y is X Y because they cannot dare exponent cannot be added because they are two different basis. So they will stay like this, x, y multiplied stay the same. We will do some more examples. So here are some more examples. Say we have eight x y divided by 2x. So again, these are unlike terms because x and x one, these are two different bases. Because in multiplication and division, we can handle unlike terms, we can simplify them. So eight divided by two is four. X will be divided by x. So x divided by x is 14 will be multiplied by one. We don't have to write per time, just writing to explain. And then y will stay the same. So the answer would be four y. The second example, XYZ it divide it by x, y. It would be easier for you to write this in a fractional form. So you can write like X Y divided by X Y. And now you can easily see that this x will be cancel out with this x. And both Y will be cancelled out because y divided by y is one, x divided by x is one. So we don't have to write 11 here because one multiplied by one is again one. The answer would be simply zed. Another example, a little bit complex. So 15 X cubed, y squared zed. I wanted to show that if something looks like very complex or difficult, it's not. If you know the basic rules, you will not feel any problem at all. Because it is mathematics. You don't have to memorize many things here. You just follow the rules. You should know that all you should understand the rule and when you are following rules, no problem how complex it looks like. Not have to worry about. So 15 x two y squared divided by three x, y, z. So again, it will be easier for you to write in a fractional form, phi x cube y square z. When you are dividing, it is always convenient to write in a Fresnel phone 3x, y, and z. Now this, now this fine. You can cancel out with 35 times. X, will cancel out with x q. You are left with x squared. X cubed root of x can be written as x multiplied by x multiplied by x divided by x. And when you cancel out this x with any of these, you will be left with x is quite similarly, this y cancel out with this y square. You will be left with y, you only, and Zed will be cancelled out. So the answer would be five. X is square y. This will be the answer. Or the next example is grantee XY minus ten YX. No, this is a minus operation. So in addition, are in subtraction, we have to check for the light terms first. So whether these XY is like term to YX. Yes. Because order of the letters doesn't matter if letters and their exponents are same. So they're like terms, order doesn't matter. So therefore this can be simplified. So 20 minus ten is ten. And x y, you can write X, Y, r, you can write y, x. Both are correct. Because both are same. Another example, 2x squared minus 3x. This is a subtraction operation. Unlike terms cannot be simplified, stays the same. So this is the answer. No simple. Further simplification is possible. Take another example. 2y squared minus two y cube. And I have seen students, they just look at two y 2y. There said no, the exponent is different. So unlike terms, stay the same. No further simplification possible. I'll take one more example only. And at the source of error for students, five AB minus BA. So b and a b are same just like Y, X and X, Y here. So they were same. So VA and VB, I've already explained. Now what about this b? It has a coefficient, implied coefficient. It's not visible. If it's not visible, it's one. So five minus one is four. You can write a, b or b, doesn't matter. So this would be the answer. So I hope this makes sense. Now here at the two points, two important points which we learned from these examples. So the first is that for addition or subtraction, only like terms can be added or subtracted. And unlike terms are carried on as it is, they stay the same. And we do not remove them. When we say they cannot be added, it means they stay the same. So if there is 5X and plus five, we cannot add them further, we cannot simplify them further. So these two terms will stay the same. So this is the thing that only we check for like terms when we are adding or subtracting. The second important point was that for multiplication and division, we can multiply any term, whether it's like or unlike. But the only thing we should be taking care of that because this multiplication will be affecting the exponent of the basis x or y, you are variables. So we will be getting those letters separately. So if 2x y, as we have already done that, is multiplied by two YZ. So we will be treating this y and y separately x and they had all these three separately. So let us multiply it as four then because there is no x there in the second term, so x will stay like this. And why they are two y's. So y multiplied by y is y e square and zed stay the same. So when we are talking about these exponents, we will treat all letters separately. These two are unlike terms. We're multiplying them. So this is the second. And the same thing happened when do IID, so far multiplication or division. There is no problem of multiplying or dividing, unlike terms, but we have to handle all letters at the variables separately. So this concludes our discussion on the operation of like and unlike terms. And you will see some problems in the quiz. And you can do there. 25. Degrees And Factors Of A Single Algebraic Term: So the next thing in this introduction to whatever glasses the degree off the algebraic terms. So if we have another debit minus five xy square so they're Diggory is the exponents value off the base so the exponent is too. So their degree of this algebraic is to let's take another example minus three xy squared. Why Cube? So the degree of this term would be the some of the exponent of both letters because letters are the base. So the some off exponents are both letters is five So the dignity of this term would be flight. So five x way has a bigly off school because if there is no exponent, it is assumed to be one. Let's take an example of and the public expression minus five X plus two. So what about this? So this algebraic expression has three terms in all of these three terms have degrees. So this there obviously Heather Digby off school this term Heather Digby off one because the expert in this one So what about Mr What is the degree of district? So if it does not have any letter in it, then the big is zero because this is a constant. This is a very simple concept. Off degree, often a generic term. No. The next concept is defectors off a term factors of a public term. So let's take an example. XY squared is a generic term, so XY Square can be written as X Marty, blind by X So these two are the factors off X So X squared has two factors. X and X. Similarly, three X is in a jab. Richter. It has two factors. It can be written as a product off three year next. So three X has two factors similarly more complex. Example. There's a 15 x why you square But this time can be return is to be market burdened by five multiplied by X. This is a multiplication sign are deployed by y and multiplied buddy. Why so all of these are the factors off this term. So these factors are very important when we will be solving these algebraic equations. So we're taking common from various algebraic terms. So when we take, you take common. So you should understand the concept of factors and we will discuss this factory ization in pre algebra section of this course. We will discuss district transition in detail and hope to find factors off numbers because these coefficients are numbers and you should be able to find the factors of this number. For example, if it's it is 51 so hard to find the factor of 5 to 1. So these type of things we will be discussing in later lectures in the pre algebra section . Just like any numbers, we can also fact arise algebraic terms. 26. Introduction to Exponents : So let us discuss about one of the most important topics in the Jabarah explanations. What are exponents? Exponents represent repeated multiplication. What should we know about exponents? Was there a couple of things we will be learning during this core during this section? About explanations and you should pay really close attention to learning all of these concepts. If you know these concepts, you may skip this section but most tutors again. I think they should take decision as well. So are we represent these exponents in different forms and what is unclaimed? More duplication. It is very important that we understand that when we put an exponent on top off are variable at an indigent. There is an underlay multiplication happening there. Next, what is the difference between power and the route? So we will be learning this also indeed your invitational exponents, then multiplication of quantities with exponents, photo Marty play two quantities which have which have exponents and then division of quantities with exponents. And then there is a concept off exponents off exponents power off the power and we will see how to tackle those type of questions and problems. And then finally, simplification of the exponents if we find ah very complex genetic expression with exponents, so hope because simplified that expression. So let us start from representation of the exponents. Exponents are represented boy and intelligent on the top, right off the variable are off the indigent, Whatever the case may be. For example, if X is a very able so exodus to the power you're in X could be an interior and could be an interview bought up, then be interviewed. Rx can be a variable so excuse to de power and it is read as extras to de power. And it means that we have to multiply X 10 times. Whatever is the value of the exes and these ex are deployed by X. I want to play by eggs and dimes. Take a simple example. We have X rays to the power three. So we have to multiply X three times eggs mark bred by eggs, one of the blade, but x no for a minute. Just compared this bint addition and multiplication. So if we have weird to read it blustery plus three, so we can achieve the same result but weren't applying three three times. So this is multiplication from addition multiplication. We are trying to simplify the process off addition, because they, for example, we have to Editor lasted 10 types three plus three plus 3 10 times. So instead off adding 10 times, we say we multiply three with 10. So this is a process. The process of March application is a simply fired form off addition where duplication is a simplified form off addition. So we have simply fight from this addition to multiplication. So in the similar fishing, we're trying to simplify multiplication with exponents. Or instead of writing X into X into X, we are putting an explosion at the toe. Everyone saying that okay, accidents to the power 10 would mean that we are more deploying x with itself 10 times x multiplied by X, my depression x 10 times. No, I'm putting all this together addition, multiplication and exponents so that you can see the difference accident toe. But we're then does not mean they're 10 is multiplied by x. No, it means that we have to march apply ex with itself 10 times so you can see the difference between addition and multiplication and explanation. So similarly, For example, if we had today. It is to the power, then here. So it for me that we have to multiply three with itself 10 times, not three multiplied by 10 but multiplied three with itself 10 times. So this is the difference I wanted to show you by putting all these three operations together. This is an operator. Exponent is an operator. But because we're not using any symbol because it is so prominent in the top off variable that no symbol is required for this operator. So this is what we represent experience. 27. ALG 30 Negative Exponents C: So now let us discuss some of the special cases of exponents. So first we will be discussing negative exponent. So one example would be x raise to the power minus m. So the easy way of remembering this is that whenever there is a negative exponent, you simply flip it over. So this is equal to one divided by x, raise to the power n. This is the easy way to solve the negative exponent problems. But let us define these in the same way. We have defined x raised to the power n. So what we do for our redefine x raised to the power n, we defined it. Exit is two. The power n means that we have to multiply x n times whatever is the number N. So this was the way we define n to the power x raised to the power n. So what we do, how can we define exit raise to the power n in the similar manner? So well, very simple. When the exponent is negative. So it means that we are multiplying the reciprocal of the base n times. So the reciprocal, the reciprocal of x is one divided by x. Based on the property of real numbers, the reciprocal F xs one divided by x. So we are multiplying by the reciprocal n times. So this is the way we can define a negative exponent in the same manner as we have defined the positive exponent. So now we have the similar type of definitions for both exit is two, the power n and x raised to the power minus n. So the easy way is when you are solving problem, you can solve in this manner. Just keep this in mind and you can solve the problems. Because when you solve this, what is this one multiplied by one, multiplied by one N times will be one. And X multiplied by X in the denominator would be x to the power n. So it means x raised to the power minus n is equal to one du or u by x to the power n. So now let us take an example. Seminal raised to the power minus p. So this is equal to one divided by seven raised to the power two is equal to one, do already by 49. So some people also suggest doing negative exponent in two steps. So the first step one is ignore the negative sign and just find out similar to the power two, which is 49. And step two is, take the reciprocal of this number, which is equal to one divided by 49. So whatever approach you find easy, straight away, you can take this minus sign to the denominator and make it positive r. You can ignore this, find the positive power and then may make it find the reciprocal of that one. So this is not, this is also true in the opposite. For example, if the question is, there's taken example. If you have to find one divide by five raised to the power minus two. Again, this is in their dominator to bring it into the numerator and change the sign of the power. So this is equal to five raised to the power. Ux reaches 25. So in boat with whenever the exponent is negative, you simply flip it over and find and Jackie to positive exponent and then multiply it as usual. 28. ALG 32 Zero Exponents And Exponents Of Negative Terms C: Navi, take another case. What if the exponent, if the exponent is one? Really simple case, but worth mentioning, x to the power one. So this is always equal to x. So just like the coefficient x has a coefficient of one as well, but we don't write it. So in the same fashion. If x raise to the power one, we don't write one, simply x. So this was the unity exponent. Now what about if the exponent is 0? X raise to the power 0. So 0. Anything, any number raised to the power 0 is always one. Command will further explain this when, when we will do the multiplication of exponents. So keep this in mind that when we were doing multiplication of exponent, I will explain, I will elaborate this further. But for the time being, you can just simply say that any, any integer, any number raised to the power 0 is one. There is one exception, reaches 0, raise to the power 0. This is not defined. This is an undefined number. And you will find it out. And we will do the multiplication. So 0 exponent is always one with an exception of 0 itself. No water vote. Exponent of a negative number. So exponent, because currently we have studied only positive numbers. When I said negative number, I mean the base is negative. So let's take an example. Minus four. Is two, the power two. So this is again, just like the body because the exponent is positive. So we will multiply minus four times minus 4x multiplied by minus four. The important point t, It is dead and it will be multiple, a negative sign. With the negative sign, we get a positive sign. And 42416. So the answer would be 16 positive. So it means that we have, in this situation, we have minus four. What about if we had minus four raised to the power three? What would be done setting? So again, multiplied three times minus four multiplied by minus four, multiplied by minus four. And these two minus will make it plus, so plus 16. And then this minus four stays there. Not this plus or minus. Plus is a minus sign as 16 into four is 64. So minus 64 would be done set. So keep in mind that then never did is a negative sign. And you have the exponent of a negative integer. You also use the sign after b's in the exponent. The easy way to remember this is easy way for negative signs. For because we are talking about negative b is, the easy way is whenever the exponent is, exponent is an even number, the sign of the desert and would be plus. And when the exponent, this was when the exponent is even. And when the exponent is or, or, then the same will be negative. A problem might be, for example, if we have to find minus two raised to the power 17. So you don't have to find what will be the sign. Just bought, we got this or just put minus sign outside and then make this positive. Today's to the power 17. And whatever the number you find on the calculator, just do the math and the answer would be minus two, raise to the power 70. But the most important thing is that you have to take sign into concentration. Sign into consideration. Sign is important for exponent, for negative exponent. So this is the summary of this section. 29. ALG 33 Perfect Square And Nth Roots C: No, I would like to mention a few other concepts which are very important when solving algebraic equations. So algebraic equation will becoming one of the next sections. But these concepts you should look into now. And if you already know these things like perfectly Squires, perfect cubes, and then the trued, you can skip this section. But I think this might be helpful for some of the students to what is perfectly square. But 50 squared is a number which we get when we multiply any other number twice. So this is a number which we get when, when, when we multiply any added number twice. So what will be the first Perfectly square? Because if you multiply two twice, we get four. And similarly, the second perfect is quite is nine, because if we multiply three twice, we get nine and then the 16 and then 2536. So whenever these number occurred in any expression. So you should think that, okay, this is a perfect square. This might help me to solve this equation, factorized this equation. And no, Similarly, I have not mentioned in the title, but there are also perfect cubes. So like the first perfect cube is it. And perfect cubes or the number which we get when we multiply that number three times, not twice but three times. So these are called perfect cubes. So we get eight when we multiply 23 times. So eight is a perfect cube. The next perfect cube is if we multiply treated three times. So we get the next perfect cue, which is 27. And then if we multiply 43 times, so we get 64. So this is the third perfect tube integer. So whenever these numbers occur, then you should understand this and you should try to make use of these numbers. And also when we are solving exponent problems are an attitude problems which will be coming next. Then these numbers will help us to simplify the expression and to simplify our problems. So it is very important for you to understand this thing and, and literal concept. We should study site by site of this perfect disquiet concept apart for QC concept. So what is N networks? And the truth is, so it is also a number. Then let route. Let's say, we say that n Let root of a number x is a number. Which when multiplied n times by itself, multiplied n times by itself, we get x. So the nth root of a number, x is a number, let say y. Let's say y is then net route, which we are trying to find. An x is a number and we are trying to find the Emmett root y of this number. So if you multiply Y n times whatever n is, if n is two, so we have to multiply Y twice to get X. If n is three, we have to multiply three times to get x. So for example, let's start from the square root. Square root of x. Square root is basically here it is two, but we only use the radical sign and we don't use this to, which is the square root. Similarly, there is a cube root of x, which is the third root. And we use three, we only omit to and no, let us say, take an example of these numbers. So if for example, instead of x, we have four and we are trying to find the nth root me the second, the square root of four. And this can also be written like this. Because I have shown you an exponential. Then whenever it is squire root R with a radical sign, the power is actually one divided by two. So we are looking for second rule. So m is equal to two here. So what number if you multiply two times, we get for the number is two. Because if you multiply two times, if we multiply two times, we get four. So it means the square root of four is two. Which is why. So otherwise too. Because if you multiply 22 times, we get four. Similarly, take a two-group, let's say 27. So take this example. And we have to find the third root and is three and is equal to two b. And this can also be written as 27 to the power one divided by three. So what number if we multiplied three times? Because we say the number is multiplied n times by itself. So if you multiply 33 times, we get 27, which is our x. So it means the cube root of 27 is three. So this was the concept of perfect squares, perfect cubes, and enter troops. So keep an eye on these type of numbers whenever these numbers occur. So you try to use them and try to simplify the situation prior to simplify the expression. 30. Multiplication of Terms Having Exponents: whether this talk of old Martin application off gums hit Having exponents so like to see it , we have already able. Okay, it is to the power looks it in and we multiply this with it. It is to the power em. So the answer would be First we have to check if the bases seem Yes, both would terms have based a only Then we can. You just rule and we can multiply them and we can simplify them otherwise, if the base is not same, then they will stay next to each other. There is no simplification. But if the basis same and the answer would be the basis and bless him, the exponents will be added. So this is the multiplication off the exponents. This is not the addition. The exponents will be added and it will be very clear to do when we take the next example to raise to the power three multiplied by food is to the power to well, the bases seem. It means this can be simply fight further. And the answer would be to raise to the power t bless cool Twitter is to the power flight. And to restore the power fire is 30 if I'm right so as you can see that they only have the basis seem Then we can just add the exponents. And for example, if you don't add the exponents then what we do to raise to the power three is two multiplied by two, multiplied by two and multiplied by putting a toe dip. Our toe is go multiplied by pool So this is it Will unwto tourist to the power five because two is being multiplied five times and according to the definition of the exponent, this is to resto bar fight. So whenever you multiply to terms with the same base, the best would be seen. You have to be careful that only if the base is seem. Because these things I know they're student maybe stop a mistake if basis seem only then veered only then veered The exponents were the explainers in multiplication We're multiplying. So for example let's say there is another case. It is to the power cool and then be this to the power three. So this time cannot be simply fight for them because the basis aboard are different. They would stay like this. How we were. Take another example. We're areas to the power three beat It is to the power to And then it is to the power minus two. No, Here the situation is that two terms have same basis and being murdered. Play. And as you know, from the properties of real number that you can change the sequence of multiplication you can put this year here and be at the end. It will not make any difference. So if I relate the problem like this, it is to the power three multiplied by either to the power minus two mark deprived by beers to the power to. So it will not make any difference to the answer. And no Mike in applied this rule on these two terms so you can see that this is equal to he is to the power. The addition off three managed to is to be plus minus two, which is three minus two multiplied will be there to the power to which is equal to area two or three minutes. Buoys area to the power one, which is here, as I explained earlier, and B to the power. Good. So this this type of Trump's can be simply fired where two terms at least two times have same basis. So we can combine them together and then simply for But they're both times have different based. Are you one of the three times and all three bases are different, then we cannot for the simplify those type of exponents. 31. Division Of Terms Having Exponents: so no, we're doing the division off the terms having explanation for deviant group to recall the negative exponents which we have just study negative exponents. So what we started did that if x rays to the power minus two, we can bring this with the denominator by changing the sign of the exponent. And similarly, if exodus to the problem minus two is in the denominator so we can bring this to the numerator by changing the sign of the exponents. Simple. So know when we're winding two terms having the same base, for example, taken example, it will be easier accidents to the public fight de worded boy X rays to the power three. So we solve this problem. Well, we just bring this term with the numerator, so we say x five. And when we bring this time for the numerator, the sign of the exponent will change actually a to the power minus three, nor some multiplication. So we can work this division into a multiplication very changing the sign of the exponents and Norway simply multiply. So what would be the answer for multiplication? We just aired the expires. So by adding for you, bless minus three, which is equal to accidentally power for U minus two, which is equal to X Esquire. Excess squared would be done, sir. So when we divide toe terms which have exponents with simply bring the denominator toe the new miniter and change the sign In other words, we can no farm earlier this to generalize this thing we say Okay, if exit is to the public and is divided by exit is too deep about em. Any other explanation could be saying But so this will be equal to X ray to the power and minus m. So the exponents off the new mediator experience of the dominator will be subjected from the exponent of the neutered. So this is just oppose it toe the multiplication In multiplication we add the exponents and division we subject the exponents and this is a whole. We proved this that actually if we bring this exponent put that illuminated the sign will change As we have already discussed in negative exponents that negative exponents stand we can work to positive exponents by bringing from numerator to denominator Are denominator to numerator. So you have derived are derived. But just explain this formula that ex If the to terms are being divided, the deserting would be the difference off the exponents. Numerous different of the numerator minus the terminator and the base will stay the same. So this is the very we do there division off the terms with exponents and no, once again we visit. Why Exit is to the power zero are any number any number there's to the power zero is always one except those you know which we have discussed earlier. So why it is so to explain this, we can write X rays to the power zero equals exodus to the power one minus one You can read zero is one minus one or two managed toward three minutes to any number. So I just put one minus one. So no, you can see that two exponents are edit together. So it means we can I can write this and number two multiplication form the X ray to public one multiplied by accident to the power minus one. Because this is what we do If two terms are being multiplied. We heard their explanations. I'm doing very worse. I haven't turned in which two exponents are being added. And I'm converting this to two terms. Exited about one and access to the board minus one. No exit is to the power one. I will keep it there. But using the negative exponents, I will take this X minus one. So the dominator and make it positive because when they were that term goes from new minuto denominator, the sign of the X one and changes. So the minus one will become plus Warren the Dominator, and you can see what is going on here. You can simply cancel these two times out. And this cancellation really deserting one Tzeitel, as explained earlier, that when you cancel these two terms in numerator and denominator, you're left with one, not zero. So therefore, this is equal This it was one. So it means excellent with the power. Zero is one because we are actually be winding X with X. So you it's really simple to explain 32. Introduction To Equations And Their Solutions: over. Next topic is solving algebraic equations. This is an introductory lecture about the solution off algebraic equation. What are fabric equations? Algebraic equation is a statement mathematical statement in which we have an equal sign on both sides of this equal sign. We have algebraic expressions, so this is one expression on the left hand side, and this is another expression on the right in sight. So any mathematical statement off this form in which we have and a public expression on one side in a diabetic expression do you that site and an equal sign He's an algebraic equation , for example, because example makes sense. So let's taken example that five X plus two equals Trent. So they have an expression on the left inside we, which we can write, left 10 site and this is a right insect. So when we say this is an algebraic equation, it means this equal sign is telling us that whatever is on the left hand side is equal to a light in sight for some value off X for some unknown X, so that if we can find that unknown X, which makes this equally in cool. If we could find that value affects, which could make this equation true then that, well, you would be the solution to this equation. So finding this X, he is basically solving the equation. Finding X So what will you off x mix this equation? Cool, because I have made this question. So I know the value off actually going toe to toe makes this equation true, because if I put five to plus two, you get through 10 plus toe his trip, so 12 equals great. So X is equal to makes this equation truth so actually occurred to to is the solution off this equation? This is not a random manipulation are some guesswork. There's no guesswork here because I made this question just to explain. You entered walls that what is the solution? But otherwise there are very sect and reading procedures to finding the solution. And this is what we will be learning in this course, that whole reach X equal toe to hard to reach here. So there are some well, procedures. Don't worry that this is a guesswork. There is no guessing algebra. So there are set procedures were really knows a few of the properties and This is what the purpose of this lecture is that I will mention to you some of the basic, an important facts about solving a tricky question. So this was the first. What is an equation? What has a solution? And the solution is also Card wrote off the equation. Never talk about a few other things before we talk about solution are before we talk about solution matters. The purpose of this lecture is to explain to you that there from things you must know before you are ever to solve the equations. And what are those things? The first thing you should know is the properties often numbers, which we learned. But our party's off numbers that distribute to property, for example, come on city property. No, the second t you should learn is ho toe open. Prentice's. You already know this starting. He's You must understand the concept off. Yes, it broke ALS. The concept of reciprocal is very important for solving equations. The 14 is you should be able to understand the types off algebraic equation. So these are the floor basic things which you should understand. Hotel open bracket. What are the receptacles and accept because include addictive and multiplication. In worse, it gets even worse off a number and the multiplication even worse. What example? They're giving worse off flavors win this fight and the march implicated in Warsaw flybys one. Depart by fight. Similarly, the additive in war Self wanted. Where to work for you. His minus one word about for you and the march implicated in Warsaw Wonder where there were five years flavor. So did that feel of the thing you should know normally talk about. What are the types off? Give Nick equations. 33. Balance Method Of Solving Algebraic Equations Normalized: so no over next topic is Bellis method of solving algebraic equations. So, as I explain in Italy here, balance matter is based on the principle that equals added to equals equals multiplied by equals are still equals. So if we have any equipment, it means that we have and a public expression on the left hand side and a number are any other algebraic expression on the right hand side. And these two expressions, the left hand side and the right hand side bought him the same value. Same numeric value. If we put the value off unknown on the left hand side, we will get the right hand side if there is only a number on the intensity and if we have illegals off unknowns on both sides and report the values of unknown. Still, the left hand side will be equal to right inside, and equipment will always be true. So this is a basic exemption. That equation is true and equation should stay troop. We should not be doing anything, which makes this equation false menace matter. Say that what they were. Mathematical operation bless minus multiplication. Our division you perform on the left hand side here on left hand side must be done under a tent site. So this will be clear if I take an example. Let's start from the previous example with you already taken this five X plus two equals True. So we know that this equation is true for X equals two. Because I made this question. If I do not perform the same operation, what will happen? So this is what I'm gonna explain. Why put X equals true? So I get five in tow. Plus two, which is Temple is too, which equals 12 which is the same is right in sight. But if we heard something on the left inside, if we had right this equation like this five X plus two minus three and I don't do anything on the right hand side, then this equation will be 10 plus two minus three. When I put X equals two, which is 12 minus three, which is nine and nine is not equal to 12. It means the equation have become false. So we don't want to do this. The question was true and we should keep it cool. So it means that this ministry we should also subject on the light inside Are we aired minus three. On the night in sight, We say we subject three are weird minus three. So if we aired minus tree on the right inside, then it will also become nine. And then the question will be nine equals nine. So similarly, this is not only for additional subjection. If you multiply, we have to multiply. So it was so easy. Then what's the point of learnings? Bella's met Well, the skill lies in the fact that what operation you should do first in orderto sold the equation. So what operation you should do first? Because if you do, if you do not do the appropriate operation at a particular stage of a question, you will be going away from the solution. You will not be going towards the solution. And this I will explain in one of the examples that hope can you do go away from the solution? So the skill off using this bennis matter when solving job Ricky question Is there work particular mathematical operation? Should I perform at this estate on one side and then I will do the same on the other side, that is not a problem, but what operation should they perform on the left inside? No, In orderto reach the solution. So know what are the tips for using this method of solution? And the first thing is Okay, so these air the tapes So these are not a must. Do you think this These are just some helpful tips. You can do it without these using these tips, but for a bigger this will make your life easy when solving algebraic equations. So the first tip is try to keep all those trams which involved unknown to the left hand side. So this is the first trip Try to keep terms airing unknown on left inside number To try to isolate the terms Having unknown why I tried twice to leave the unknown Because ultimately what we need is X equals something So, for example, exit over unknown So activity over goalies toe find X equals something We don't want anything to be multiplied by X so therefore we have to isolate ing's So try to keep the terms having unknown on the left hand side Why let the left hand side? Because normally we go from left to right We read English from left to right sold operations from left to right. So it's easy to keep all their terms x so that we can reach X because something otherwise we will reach like this five equals X. Then X will be on the right and said, If we keep all terms on the legs right inside, so you can keep them is on either of the size X all the extremes on left hand side at the right hand side effects is your unknown are why is that your unknown? You can keep all white terms on the left hand side are alway terms on the right hand side and all numbers without these Xer way. Whatever is your unknown variable on the other side. But the best approach is to keep the variable terms on the left hand side and the numbers on the right inside, because using examples makes it easy for students to understand. Let's take one example five X equals 15 and into that example five x minus three quarts trail and in that one x minus two multiplied by five equals five. So I'm taking these example side by side so so that I could explain a few of the things which I want to explain here, Keeping in mind the first thing that all the extremes should be on the left hand side. So this is only one term. So this is the most simplest case. And normally this is the last stage of over solving the question. When we when we have sold the question at the very last stage, we end up something like this. So five X equals 15. No, we want to get rid of this fight because you want to actually eight X. So the best way to do is no. We know that. What oppression we should do. Division. You should be white. This is my fight. So five X divided by five. And in order to balance this, we do it on the second side. 15 to order by flight. No, you can cancel this five, Betis five And this five goes three times. So we get X equals two b. This is our solution. Very simple, because it was a very simple example. The second example. So what we should do here? Should we remove this term? Ministry I should we remove this five? What we should remove first. So when there are two terms No, the there two times one has six. One does not tell wakes. So when there are two times So you want to get rid of the second time with doesn't have X So in order to get it off minus three What we do five X minus three. We had plus tree putting move this time and this blustery should be added on the other side . So here we get five X because three Ministry zero our ministry plus three zero on the right . Inside we get 15. So no, we remove this five as we have done in the last example. So X equals to be just like this. So what about this third case? Should we perform this five division first? Are this addition first? How should we proceed in this example so well for the big nerves In this case, there are two matters doing this. But first I will explain the bigness matter. So if there is a bracket, this is a break it opening here and closing years. This forms a group over and there is a bracket, the easiest approaches. This is the first matter, which is the easiest approach. If there is one break, it are. If there are two brackets open the brackets open The breakfast first opened. Uh, records, Part parent is whatever. If I opened the breakage, I get five X minus 10. Ik was flavor. We are not not doing anything on the left hand side. Extra were just opening the bracket using the distributive property of numbers. Remember in Toby plus E waas a B plus Easy. You might have read this with this. Plus you multiplied this with this. So we're doing the same thing? No, we have two times. No, it is like this question if you compare this with this very similar. So we remove this term by adding plus 10 flavor mix. My understand last 10 and we do the same plus 10 on the United States. This will be removed. Five X equals 15. No, this is just like this question. So know what we do? Do you worried by five And the word? But if I on the other side X equals trees down So So this is the easiest matter If you have a group, if you have apprentices just opened the parentheses and solve it. But there is another method which I would also want to highlight and effect, which is that if you look at here for you, I write related the third part again here. And if you look here, it's only wonder there are not two terms because of this grouping symbol. This is one time so this can be treated just like this here. X was being were deployed by something. Here are Group A group containing X. The X minus two is a Group X managed to is a group containing X is multiplied by five and there is only one there. So here we can also divide first, we can also first divide. This is the thing you should be learning that. What operation should I do here first? So you can also divide first because only one term so it can be considered a case like first case. So we can divide by five on both sides and we will get for you X minus two. Do everybody five equals flight diverted away for you and you can cancel this out. Cancel this out. An X minus two equals one. Remember when they can sell old one is remaining not zero. So no, we have to terms So we can remove these two by adding two on the left hand side and adding to on the right hand side. So x on the left inside and three undulating say so if there is only one term, you can do it. Israel Another example extorted by three bless X equals 20. So what should we do here? So over first tip, Waas tried to keep all terms having X on the left hand side and all terms all numbers on the right and say so if you don't follow this tip and you proceed like we did in the second part of the previous examples where there was a second time But I first do the wrong way. So this is the wrong way off doing this. For example, you decided to remove this term first so you can remove this term by minus X o X plus 3 October three plus X. If you remove this, want to remove this? You just aired minor sex. You can remove this term and you have to do the same on the other side. So you get 20 minus six. So the time is removed from the left hand side and you were left with extra worry about three equals 20 minus X. So what happened? No, you have removed XT m one of the terms on the left and say, but the term has moved to the right inside, within up, What you're saying? So you have gone away from the solution and instead of going closer to the solution because you just looked at the second time and you try to remove this But you forgot that all the X terms should be kept on the left hand side and all the numbers should be gone under the right hand side. So this is a time which has X so we don't have to move this to the other state. Otherwise, it would be a problem for us. No, we have gone away from restoration. We have to remove this from this side. And if you removed is it will come again on this site. So we have to put all X times together in all numbers together. So what we do have to do or ex terms together, and then all numbers together. So this is our first deep. So know what we do here? I just come from here again. So what I can do in this is I can take X common from these two terms. So what will be left inside? One divided by three Bless one. And if you don't know how to take common, you can wash that letter again. Taking common. There's a lecture Really lecture. You can wash their because you will be taking common again and again. So I wonder whatever three plus one and on the right side is 20. So taking common, doesn't it saying anything? Just if you the distribute to property you can multiply these two and you will reach there extorted by three. Then you Marty plea one with X and you will become here. So you were not changing anything. So no x is common. I just sold the freshen inside. Wonder where to buy three plus one. And if you don't know how to solve this this friction then you watched that we do tradition off frictions and I just saw this here So x so I will take the denominator three. So one plus three equals 20 x multiplied by four, divided by three equals 20 and no, we have one term on the left inside, which has eggs and one number on the right inside. So know what we can do is we want to get rid of this four by three. So the easiest place to multiply this bid three by four so that all cancers up. But to balance this, we have to multiply three by four on the right hand side. No, you can cancel this out for with four three with three and this four goes five times. So on the left inside we have only X X equals five more depressed but trees 15. So this is our answer. So in this example, where to learn these? If there are two terms and both have X, then you don't get rid of the second time because you want all ex terms on the left inside and all numbers on the right hand side. And then in the last step, you want to isolate X. When you want to combine or less terms, you had to take the common taking common. So this is your thinking process so I can take or text or Lex terms on the left hand side. Ho, Can I take common? How can I combine numbers? How can I simplify fractions? So all of these, we'll be coming again and again when you will be solving agility. Question. It is important that you do all the homework and you do all the quiz questions. And if you feel any difficulty in solving fractions or anything else, you must watch the video in pre algebra section of discourse that will give you confidence in solving corrections. And that conservatively Sprinkles. Because here, what we're doing is we're multiplying. There is a reciprocal afford by three in order to eliminate for my three from X So these type of things, if you find any difficulty, you go backward. That video all home what you should do for soloing equation. Because soloing equation is the major part of the algebra and in practical life, you have to formulate the problem and then solve the problem. So I will put the solution to the homework in Resource Is, and also I will try toe make a video about the quiz questions, and this would you will be uploaded after the course later. Sometime. There are different types off in question, which we will be dealing, and we will be practicing a lot about solving equations. So make sure that you practice all homework and do all the quizzes. If you find any difficulty you can contact, and if you're very question you can ask, I will be happy to answer your questions. 34. Reversing Method Of Solving Algebraic Equations: so the next topic is reversing method of solving a jevric equally in. So before we do this method, there are two things which I would like to highlight the first days you should be able to recognize one term and many times it looks very simple. But believe me, there are a lot of us doing mistake in recognizing whether it's a one time. There are many terms, so this is the first thing. For example, if we have five X plus, Stinney is equal to two. So here there are two terms. Wherever there is a sign, it means there are two terms. No. Five X is faster and blustery is second terms. If I make this a group and put up three outside, no, this whole becomes wonder because no, it's a group and group means that apprentices are starting at this point and ending at this point, so it makes a group, and when one group is multiplied by another time, still it stays one term, so you should be able to understand where there's the one termite. There are two terms, so this is the first thing, and the second thing is the concept off it is separate girls and in worse had it even worse . So if we have a number three, do whatever a fight, so it's reciprocal. Would be five divided by three, which is also called Mart. Implicated in worst and it's negative in worst would be minus three, but flight minus three before we got it has a plus sign here. Expert implicit Blessed sign, which is not shown. So this will be additive in worse and this will be its reciprocal. These two thing you should understand and you already know these things. And if you don't know this, please watch the lecture about two separate cars and in worse. And once you are comfortable with these two concepts, then you are ready to understand the worst thing matter of solving equations. No, the question is why we're using this mess. What we're trying to achieve here is we're trying to just save save some time by skipping one step, and I will explain this in a minute. So when there are many steps in the questions, you can save a lot of time. For example, we have a question like so this is an example, and I will explain this by doing this in balance mattered and in the worst thing matters. So you can understand what is the difference? So you takes minus 10 equals 30. So in Venice mattered because there are two terms. So we want to eliminate this term minus 10. So we will act plus 10. So it takes my understand plus 10. And to Bella State, we have had lust and on the other side, this is what we do. So after the next to step eight, X equals 40. So this is the way we do now. What we do in reversing matinees that we see that let's split pea put the same question. It x minus 10 equal 30. So what we do in reversing batteries, we instead off trying to remove this term? My understand read Take this time to the other side of the equation. So this is this is one side and this equation said this left turn side and this equation sign distinguishes the two sides that this is right inside. So instead of eliminating this term, we take this term we take this to the other side and replace this with its sedative in worse So what we do? The next step would be eight x than equal sign and 30 and then, my understand when goes to the other side it becomes plus 10 the additive in worst. So we take the additive in worst of this on the other side because this time is being added . And if it was being multiplied by something then we will be doing the mart implicated and worst that a separate girl of this term. So no, we have saved this This and we're state to the next step. Which is it? X equals 40. The next step is no. We have a text. So we will divide here, buy it on both sides and then we will simplify X equals fight here We don't do this. We simply take this age and divided on the other side. 40 do worry. But yet because it is being multiplied on this side. So no, we divide it on the other side because the reciprocal of it is one to everybody it so if you multiply wonder where the way yet on the other side and we again get X equals five. So in this way we try to make the process a little quicker. So when we do some more example, you will be family, and you will be fluent with this matter. So let's take another example two X plus three minus four equals 10. This is one term. This is one term because of this grouping sign to interacts plus trees. One term and minus four is the second term. So what we do in this case, we take this time to the other side of the equation and instead a minus for rewrite plus four. So the next step would be we can open this bracket at this step. But I'm just trying to explain this so I keep it like this. So one step at a time. 10 plus four No, we have to. X blessed six equals 14. No, we have to terms here. So in two terms, So we take this time to do the site. Sirbu eggs equals 14 minus six. Two. X equals it. No X equals who is being multiplied on this side. So we will divide it. We can say that we will multiply the reciprocal of this number on the other side. And we take this to do the same. Read equipment brings X equals four. So this will be the solution. So you see, it's more simple. It's more efficient method of solving algebraic equations. Another example, though this example has variable on both sides. Let's take six x minus tan. He was four eggs plus two. So one thing you should keep in mind that when using when solving question using, reversing mattered, you should always use one step at a time because it looks very simple. Toe add steps. But if you try to add too many steps, then you will do mistakes. So one step at a time. So let's take this example and see how we proceed. So the first thing is that we want all extremes on the left hand side. So we want to bring this for X on this side. So one step at a time so 66 and we can combine extends together No problem. So I will write this minus four x before 10 so that we can combine. This equals two so four x When comes to this side, it becomes minus for X, The additive in worse off four X is minus for X So we take minus forex on this site. So whenever we take time from one side to the other, we take the additive in Worse if the term is being added. So six X minus four x again, we can be a common here we can take excrement and six minus four minus 10 equals two six months for these tools. So it is two weeks. The works, minus 10 equals two. No, we can take this 10 on the other side. My understand, to get the site So this will become blasting. So two X equals to bless them, which is 12. So two X equals 12. I'm just writing this extra step toe the mustard to you. Otherwise you can skip this. So know this. Who is being multiplied here. So we will Marty play. I will take this to the other side and we will multiply with that A separate girl of prohibition under whatever to out. In other words, we can say that who is being multiplied? We will do white toe and we take food and we take this. Put that aside, we just do I buy two. So as tech seek was well, do everybody toe reaches six nor picking of them example, This is more complex. It has two groups, one is in a group and one is outer group So five and we started out a group and there are two types of group you normally use curly brackets and parentis is so then four inside and then in a group which has X plus one and then here Brooke Group clothes and then out. A group also closes here, then minus twigs equals 30 plus edicts. So here we have X terms on both sides and also two groups. One is in the group. So in this case, we start from the Intergroup, always start from the other group and then go out. So from inner to out, remember the tip for the bigness when their brackets toe easiest ways toe first, open the brackets. So I will just keep this outer five here and I just opened the inner group. So which is four X plus four minus two weeks is equal to 30 plus eight x so, no, we open again. So it becomes 20 X plus 20 minus two weeks. And at this stage, I can save time by bringing this here, Tex. But this side of equation and depressing it. Expert minus eight X minus eight X equals 30. Why can combine all these ex together and I can write plenty X minus two weeks minus etix and I can bring this 20 in. This is step. I can bring this to the other side and change it to minus 20. So 30 minus 20. No, I can pick Exc woman from these and I will be left with 20 minus two minus it. And if you don't know to take home and take the lecture It was a lecturer with you for taking Commons and 30 minutes 20 sten for 20 minus two minus eighties 10 through 10 X equals 10 and no excess being multiplied. Here, take this to the other side and divided. It gives us X equal one. So it was looking very complex and it was having values on both of every Abels on both sides. But it proved to be very simple and very quick and efficient. So using that reversing matter, you can save a lot of time. Simply take the time to the other size and change the sign are. The term is being multiplied. You divide it. If something is being divided on the left inside, you take it to the other side and multiply. I hope this will explain to you hard to use a pressing matter to sort of the idea. Ricky Williams. No, Only if you practice this a lot, then you will same save time because practice makes you perfect. So there will be a lot of questions in homework and in quizzes. I will not ask questions like this because this is taking too much time. But this double questions will be in homework and in cuisine. There will be some simple conceptual question, which you should also attempt. So this was it for the worst thing matter. And I hope no, you will be able to sort all the homework questions. 35. What Is Factorization And What Skills You Need: so the next topic is basics affect realization. What are we thinking when we're doing for actualization? What skills do we need before we attempt for transition? So let's dive in boot. So before you attempt a tradition, as I mentioned earlier, there are a few things you must master. The first thing is hopeful. Find highest common factor at the greatest common factor. So this you should muster before attempting factories issue hard to find DCF. If you don't know how to find these here for two numbers and more numbers, whether the number they're smaller big, then you should not attempt for a transition because this is the essential requirement for doing factory ization. The second important thing is hold today with the exponents hope to handle exponents. Exponents are very important in algebra and if you are not familiar, if we're not very company comfortable with the exponents, then you should first watch that lecture direction. Actually, a couple of lectures about exponents and then you can come back and Vortis lecture and the tech 13 is hotel a common. This looks very easy, but there are few things. For example, the most important thing is hope to handle Sines. If you are not very comfortable with science, then you should watch the video multiplication or division of signed numbers and many other places. I have discussed this thing. For example, if there are two numbers minus flavor plus 25 So what is common? Obviously five. So there are two ways to take 51 is you can take plus five common and what will be insight inside it will be minus one plus five. The second way of taking companies take minus five comma. What will be inside? Because you have taken minus same common. So all the signs inside the bracket will change. So then one and minus five will be inside the bracket. So this is what I mean by signs because this is what we were doing in fact, realization. And the fourth thing is one of the basic thing, which is that you should be master of handling fractions before you do factory ization. Because many times what happened when we take something common, there is a fraction inside the bracket and you should be familiar with taking Commons. Infractions are involved. These are the few things if you have already done this thing and you have done the quizzes and practice questions. Then you can board this. We do. Otherwise it would be better for you to first watch these reviews on these topics. And then you come back and do the factory ization. Okay, so, no, what is factories issue. Okay, you know what is factory ization? Perfect transition is putting some algebraic expression into a product form from addition form our subjection form toe a product for fact transition is opposite off opening brackets . So if you have a bracket like five x minus tool. So the baby opened this packet days we multiplied five with X and then we multiplied five with minus two using the distributive property of numbers and we get five x minus 10. So know what If we were given this term and we had to bring this into this form So this will be factory ization? No, this is an algebraic expression which is in edited form. A dutiful means that comes are being edit are subjected and we have to bring this added our addition form in tow a product for here too. Trump's air being multiplied. So this protests car factories issue, so there could be two terms. A very simple case. This is the simplest case off a position with only two terms, then three terms and then four terms. So depending on the situation, sometimes four terms is an easier case to handle them. The three terms, because in three terms we have to modify the terms while in four times. Sometimes we can straight away take Commons. So the easiest case would be when there are only two terms and in most cases than the four term case will be the easier one. And then the three terms. So these air, the number of their could be many number of times. But 1st 3 situations are these three, so this is increasing level of difficulty. 36. ALG 42 Terms Used In Polynomials C: Our next topic is polynomials and related concepts. The word polynomials as two components, polynomials. So poly means many in Greek language and monomials mean names. So in algebra, this means terms. So any expression which has some algebraic terms, some finite number of algebraic terms is a polynomial. Let us take an example. 15. Excellent is to the power seven minus four x to the power six, but less phi u x 5y minus three. So this is a polynomial because it has four terms. First term, second term, third term, and the fourth term. And as you know from the last lecture, that the term has three components. So let's take this term. It has three components. It has a coefficient which is 15. It has a variable, our base, which is x, and it has an exponent which is seven. So note that we are talking about polynomials. Why this is important for us to keep this in mind that any term has these three components. Because there are some related concepts which involve the understanding of exponential. So the first thing you should note here is that in any polynomial, the exponents must be non-negative integers. So if I have an expression, 13 X4 plus root x, so this is not a polynomial. Because xth root, root x means x raise to the power one. Do are the B2, which is not an integer, which is a rational number, which is a fraction. So this cannot be a polynomial. This is the first thing you should know that a polynomial must have all of its Oliver exponents as non negative integers. So what about this minus three is the generic term. What is the base? If you recall your exponent lecture, any number raised to the power 0 is one. So it means x raised to the power 0 is one. So I can write this minus three as minus three x raise to the power 0, though it has a base and it has exponent. So it means this is also polynomial return. In this lecture, you learned that a polynomial is an expedition and algebraic expression which has some terms. Times could be 123 or four, but finite number of gambling cuz we thousands of transport, finite number of terms. And you were it can has a numbers. Like 344 any number. Now we are going to discuss some of the related concepts. So the first thing is I am going to number these things, but I may skip any number. So the first concept is monomials. So monomials are the polynomials with only one term. So if I have an algebraic term, 5X is quiet. So this is our monomial because it is an, it is a polynomial with only one term, so it is a monomial. The second concept is binomials together you can say one way of describing the types of polynomials. There are few ways to describe the types of polynomial. This is one way which is based on the number of terms. So we are classifying the polynomials based on the number of terms. But there are other ways of classification as well, which I am going to explain next, but by no means are the polynomials which have two terms. 5x cubed plus five is a binomial. A plus b is a binomial. C plus zeta is a binomial. Any letter, any two terms. Now, you might have noticed that this has only x in it. But these two, have two letters to variables. And this is another way of classifying the polynomials. And that is univariate polynomial. Univariate polynomials, a polynomial which has only one variable. It means all the terms are in one-variable, five, x squared minus 3x plus five. Only x, 2A plus two, only a. So these are all univariate polynomials. No, the multivariate polynomials are the polynomials which have more than one variables, such as two x square y cube plus five x y square minus two x y plus three. This is a multivariate polynomial because it has two variables, x and y, X, Y. So if even one of the term has two variables, for example, we have a term, phi x cubed y plus 2x plus 3y minus three, for example. So only one term has two variables, Y1, then it will be a multivariate polynomial. Because in this term we can say that y is 0. Because y 0 is one. In this term we can say x is 0. X has an exponent of 0, so it is still a polynomial, multivariate polynomial. So this is another way of classifying the polynomial. Which is based on the number of variables. This way is classified the polynomial based on the number of terms. So you should be just familiar with these two things that diesel here we are classifying based on number of terms. And in this case, we are classifying the polynomial based on number of variables. So you just should know these terms. So the next related concept, number five is the degree of the polynomial. So again, an example, it can exemple five x 3y Fosdick single-variable polynomial 5X cubed minus 2x squared plus two. So the degree of the polynomial is the maximum exponent in any of the terms. So this term has an exponent of three. This term has an exponent of two. This term has an exponent of 0. So the maximum exponent in any of the terms of polynomial is three. So this is the degree of the polynomial. Maximum exponent. So this was the case of US. Single variate are univariate polynomial. Know if we have a polynomial in two variables, multivariate polynomial. So let us take for x six y plus X3, X4, Y4 plus phi x y minus two. So in this case, when the polynomial is a multivariate polynomial, having finite degree, we add the exponents abort variable. We add these two exponents, these two exponents, these two exponentials. And what is the value of this expert total exponential of this term? Six plus one equals 74 plus 48112. And here both has 0 exponent, so 0 plus 0 is 0. So the maximum exponent is it. So the degree of the polynomial is it? Degree of this polynomial will be eight. If you look it. If you look it carelessly, you might think that the degree is six because it looks six is the largest exponent, but actually it is not. Then there are two variables. You have to add the exponents in all the terms of both variables, and then you find the maximum. So this is how you find the degree of the polynomial. Sometimes people confuse this degree with the order of the polynomial order. But this is not a concept for polynomials. This is mostly me using calculus for differential equations. But sometimes people use this degree as the order of the polynomial, where it's up to you, not the next concept is leading coefficient of our polynomial. So if we have a polynomial, five x cube minus four, x is square plus phi. So far, in order to find the leading coefficient, first, we find that term with the highest exponent, the largest exponent. So this is the term with the largest exponent because it's a single variant polynomial. So trees that term with the largest exponent. So the coefficient of this term is the leading coefficient. So if we write a polynomial like minus 2x plus five X4 minus 3x squared. Although this minus two looks like a leading coefficient when we write this like this. But actually the leading coefficient is the coefficient of the largest of the term having the largest exponent. So this is the term having the largest exponent for. So the leading coefficient would be five. This is the one source of error sometime. And the second is that if I write another, take another example. Minus phi x cubed plus five, X4 minus 3x five. Now in this case, the term with the highest, the largest, the term with the largest exponent is five. So what is the leading coefficient? The leading coefficient of this term is minus three. So you must take into account the sign. No. There are few examples which are not polynomials. For example, minus five x squared to the power six plus two divided by x minus five. This is not a polynomial. Why this is not a polynomial? Because for all trans of a polynomial, the exponent must be a non-negative integer, non-negative integer. If you look at the second term, this term is basically to x raise to the power minus one. And this is a negative integer. So therefore this is not a polynomial. First example, one, take another example. 7x square plus rho dx. This is not a polynomial because of this term. Root x means x raise to the power half. So this is positive but not integer. So we're looking for integer, non-negative integer. So this is non-negative, but not integer is a rational number, is a fractional number. So therefore, this is not a polynomial. You will find these type of questions in quizzes. So these were some of the related things I wanted to highlight about polynomials. 37. Method of Factorization of Polynomials: So we have talked about 40 Nahmias with two terms and with four turns, nor were discussing the party no meals with three times heart if actress party normals with three terms. So these type of pollen our means are also card trial armies in the similar fashion as we call the single temple normalised mono MIAs and part number with two terms because them by normal, so in the same fashion, with three times we call them, try no meals. Take any example Excess square plus five x Class six. So this is are, you know, very good party nominee, which is trying no meal. And there is only one variable, which is X. So when we're fact arising these type off 40 normals, we cannot make groups as we did in with four terms because in four times you we were able to make two times together and take something common from two terms and take something common from other two terms, and later we took a binomial common from ditch. But we cannot do the same thing here because we have three terms and there is nothing common in between these three. So in these type off try Nahmias. We try toe split the center, Mr five X into two terms. So if you make them from 3 to 4 and then we try to get something common. So there is a mattered off splitting this middle term so that we could actually get common from those four times. So what is that matter? So the matter effect rising. Try normal. Is that you always compare distraught. Try normal with a standard form. E x is quiet plus b x plus e But this is the standard for So if you compare these two, you will notice that it is the coefficient of X squared, which is one So it is one and B is five and sees six sees a constant. So first you have to return. What are the values off A B and C and then the matter which I am going to explain. It's also card a C matter. So what we do here is we multiply. It would see so one into six equals six and then we that I don't be whatever is the value of B B equals fight. No, we have to numbers you Sequels six and big was fighting and know what we have to do is we have to find two numbers such that their product is easy and there's some equals B. So we have to find these two numbers. And then we split this term. Using those two numbers, we will split into those two numbers so hard to find these numbers. Think about the factors of 61 through 36. These are the factors of six which we get from a see out of these numbers, which to number Marty pluck it six. And when we air them, we get five. So you can easily figure out that if we multiply these two numbers two and three. So we get six and we're 20. We get five, so we will split this five x into two x and three x. So once we have found these numbers, we will spread the middle term into these numbers of every rewrite of original equation X x squared plus two x bless three eggs plus six and no, if we take common, what happens? We take X from 1st 2 terms and we're left with X plus two and we take common plus three with second to come. Pick only number is common between third and fourth term. So we are left with X Plus two and no, you can see that this X plus two which is a binomial. We can take this binomial as a common because this by normal is just acting like a single letter. So this is wonder they already mentioned that must be able to identify their whether it's wantem are not so one time. Use the other terms These these two times have expressed to common because it just like this desk exes multiplied by something which is X Plus two. And the second time has three multiplied by something which is seem so we can take this thing common so nor taking explains to common when we take all this is a binomial Then we take this binomial common. We're left with X in the foster and plus three in the second and were able to fit tries. This these are the factors off actual squared plus five x plus six. So let us take another example six x x squared minus five eggs minus four. Oh, compare this with excess square less me X, Let's see and we find that equals six Big was minus fight. We have to consider the sign of the coefficient. Can seek Waas minus four. So our is minus 24 and our B is minus five. No, we have to find two numbers. If Marty plied, we get minus 24. And if added, we get five. Sign is the main reason off making errors in this procedure. So, no, I am showing you how you decide what will be the sign of those numbers. So suppose we have two numbers. One number is any number year. And the other number is is this we have to find, but because there are signs involved in here. So there is a matter you can easy to determine what will with a sign of these numbers. So how are you returning the sign of these numbers? No. If you look carefully that the sign of the product is minus, the sign of the product is minus. So then you multiply two numbers and you have to get a minus sign. So what will you have? What will be the sign of the numbers? One must be positive, and one must be negative. One must report it to, and one number must have a negative sign. And only then you will get this minus sign here when you even might play those two numbers . It doesn't matter whether you put the negative number first of the positive number first. The only thing important is that one number will have a positive sign and the one number will have a negative sign. No, If you look at the sign off B, the sign of B is negative. And what this tells us is because when we will add those two numbers when we add these two numbers, we will get a negative sign. So it means the larger number will have the negative sign. So you will put the larger number here larger of those number, we'll have a negative sign. So no, we have decided that we really be looking for two numbers. If we multiplying, we get 24. And if we add, we get five. So let's talk about the factors of 24 because 24 is the product of those two numbers. So what are the factors? That 24 1 2 34 6 it 12 and 24 so no. Look for two numbers in these factors. Whether you right here are you just doing your head Don't matter. So what do numbers? If we act, we get five and we for multiply. We get 24 1 pair will be this 11 into 24. Because the both have different signs. So from 24 to 1, we will only get 23. Okay, No, look at the next one toe into 12. It also 24. The product will be 24. But if he had 2112 and one has a different same. So we will get then. So this is not over number? Because we need five. This is our target. When we add both number, we should be able to get five. So, no, the next number next period is three more deployed by eight. So if you multiply three and yet we get 24. And if we heard three and it, we can get five because both will have different signs and the sign of the larger number will be negative. So we got over numbers. The numbers are two d and it so these are ever magic numbers and the large It has the negative sign because we have already checked this. So minus eight and plus tree and both will have different signs. So no, the same problem is solved. So this is the main source off error. So no over numbers are t and minus it. And no, we split this minus five x in tow, three eggs and minus eight x So the next step would be blessed to the eggs you can write plus minus eight x first, No problem minus eight X and minus four. You see, we're not changing the original equation because if you simplify these two, you will get minus five X. So we have just identified a way to split this term so that we could fact arise this and know you can see that we have a number common in X common in first to and also a number common in 1st 2 So what is the number between these two terms, which we can take common? 33 is the greatest common factor, and X is the letter. The minimum exponent of X is one, so we can take three x common. The same technique which we are using from the beginning that what is the Jeezy? What is the greatest common factor? Common between two numbers. And what is the minimum exponents so vividly common to the X from the 1st 2 And inside the record, we will get Truex Bless. But I was never too. And think one Then, from next to numbers, we take the negative sign. Common negative. And eight and four. What is common here? What is the greatest common factor between eight and four? Four. So we can take four common. There is no X. In the second terms. We cannot take excrement. So we get Wicks. The sign of the storm will change because we're taking minus common. So the same time next time we'll also change and we get one. And no, If you look here, you will notice that we have. This is one time and this is second. And in both of these terms, we have this by normal common. This is a binomial apart, a normal with two terms. So we have this binomial common, so we can take this by normal common. So this will be two X plus one and what will be left here three years minus four. So no record this one product form, which is over answer because we are able to can work that open form into a product form. So this is the factory ization. So what we're doing is up. Was it off opening bracket? So if you open these brackets, you will get this value. So this is all if it tries that rhino MIAs And no, I would I want you to practice the scientific first. So no, we will do few example and we will just decide the sign of the numbers so that you can have some practice off making the correct signs of those numbers because finding number is not very hard. Putting sign on those number is really sometimes tricky. So we will do some practice on how to put science in the next review. 38. Factorization Of Polynomials Using Identities: So there is another method off transition of 40 Nahmias, which is based on mathematical identities. So what are mathematical identities? These are the equations which are true for every value off the variables they have. So whatever value you put for X and y, this equation will always be true. So these type of equations are called mathematical identities. Hope used these in factory ization. We will get a poor normally in the form which is on the right hand side of this identity. So what these identities telling us that if some equation is off the form off X square plus two x y plus y squared. So this can be written as explains why holy square. So this is a perfect square off X plus y. So if you open up this X plus where the only square means if you multiply X plus y twice so you will get first somewhere to play this X with X and why So you get X X quiet Last ex wife, then you worked murder played this way with this X and this way. So you get what you x plus why you square So these two water century the same thing over the next where your ex wife. So this equals excess square plus two x y plus y squared. So it means if we get an equation in this form so we can state of a right dirty question into a place holder square. This is a factor form. This is a product off to trim, so this is affected for so in questions. If you can figure out that Upali novel that apart normal given to you is in the standard form off X plus when all is quiet so you can state away right into this and this X plus way doesn't matter whether it's a plus b r x plus y, for example, the same identity can be written as a plus B. Holy Squared would be equal. Do a square plus two a B plus B squared. So this doesn't matter whether it's X are white if there are two very Ebel's, So the square of the two variables is always equal to the square of the first variable, plus two times the first variable taken variable, plus the square of the second variable. So this is the video. Understand these identities because what happened some time you will get question in the form of X plus y, and then you can substitute a plus being toe those values and then you can use the identity in the form of a plus B. So this happens all the time. So similarly, we haven't identity for difference of squares. This is guard difference of squares, but norther. There is no identity for some off squares. There is no formula for X squared plus y squared. We have only formulas for X squared minus y squared X Q plus waste to annex. Humans work you. So this is car some off cubes, the cube of the first variable and the cube of the second variable. Because these are these identities involves square off the terms and cube of the terms. To be able to use these identities, you should also be ableto pick the perfectly square numbers and the perfect You've numbers . So which numbers are perfectly Squires, in which number, a perfect cubes are starting from 24 is a perfectly square, and it is a perfect cube off to so. Similarly, for three we have nine and 27 so whenever you see number 27 you should be able tow. I realized there 27 in the tube off three not only to three, Bert for square values. I would recommend that upto the value off 20 you should be able to recognized the square value. These are the perfect square. And for cubes, maybe up to 10 you should be able tow. Understand that 64 is the Cuba four and similarly five has a cube of 1 25 six has a cuba to 16 immediately when you see 216 usual realized that this is the group of six. So these fuel numbers, you should be able to pick that This is a perfectly square, A perfect too. Otherwise, you won't be ableto transform your given problem into one of these identities. So it will be difficult for you if you don't realize that cube are This is a square. One more thing, which you need in order to solve such a question would be there. You should understand that x six for example, can Britain is X cube is to de power the square and can also be return as extra square raised to the power three because power of the power always Marty polite. So sometimes you have to transform your question in this In this form from x 62 x, Cuba is to the Power Square to be able to use this identity, for example, because this is a difference of squares we don't have any is some of the square. So we have to use this sometimes. So if there is a situation where you have to modify these exponents, so you should we ever do this similarly infractions. For example, I take one example for a Cuba you X Q minus one to worry about 27. No, you can see that this 27 is a group of three. So it means that in order to use one of these identities, for example, this identity is similar to X cubed minus two off something. So this is a Cuba off one door by 31 by three. So this can be written as one by three with Cuba. So here we're using one of the index laws that something you order by something raised to the Power Cube equals one cube. The word it by three coupe. So basically, we first realized that one is a perfect Cuba, of course, and three is a perfect to about 27 is a perfect group of three and one is a perfect tube. And then we transformed this into one word about three whole cube to be able to use this identity. So then we can realize that this excess our first variable and wonder what about three is our second variable. So then we can put in this equation and we confected this polynomial were using these identities. You will always be going from right side to left side. So you will get an equation one of these forms and you will transform it into affected form because all these left inside is a affected form off these equations. Because all this left inside our defected phone and you can say that the right inside our like open brackets. So this is the open bracket form and left Concert is a separate form, so normal normally does a few examples off each of these cases. So you will better understand or produce these mathematic colored introduced affect tried polynomial 39. Synthetic Division:A Factorization method when one root is known.: Okay, Now we do the synthetic division of Pollen Nahmias. So let's do the same example which we have done using their division so that you can see that. What is the difference? How much time you can save? So we take the same effect on the same example. So our task waas to fit. Rise six X cube minus five x is quite minus forex. Blustery when one factor was known and which waas X minus one So X minus one is effective Incented division. We only use the coefficients off this polynomial and this number, which is the part of the factor. So we used this where we do that first port the coefficients of this party nominee with a little bit of distance so that you can easily do the synthetic division. And then we put one old site this one If X plus one is effective, then we put minus one dear, we put minus one If X minus two is affected, we put so there. If X plus two is effective, we put minus two there. So X man is one of the factors that we put one. Put one here so I know what we do is we put a line and we keep some distance. Yes, we can write numbers. So we bring first number state of a dome six 10 V mart applied This one with this six you might have played and reported Did so six multiplied way one is six. So we poor 600 This minus five? No, we add these two don't subject here we aired these two Su minus five and six. When added together we get one. So we put one here. No, again. What we do? We multiplied this one with this one and we put at a diagonal position. This is a Bagnall position. So one more depressed by one is one. Then we aired. These two were adding here. Then we aired these two. You're not subjecting. This is the mistake sometimes to undo So minus 41 When added we get minus three. No, a game multiplied This one with minus three and put it here So it will be minus three. No, it again. And three ministries zero. And this is what was expected. Because if x minus one is affected, then the remainder should be zero. This is the remainder so know what we did is instead off. I was explaining so there I was a little slow and you will do this. You will do very quickly. So you save a lot of time. You don't have to write this x x x x Again and again you simply use the coefficient and divide these two party Nahmias. So what is the desert? What is the degree of the polynomial? As I already explained, ex tree was the degree and we were the wording with the degree of one. So what? We will degree off the desert do so? No we bring. These are the coefficient of over portioned polynomial number the portion which we get when we do our two numbers we get the caution. So these are the coefficient of offer caution polynomial with one less degree. So instead of trees. So this six is the coefficient of excess square six Xs square. This is the coefficient of X plus X because it's one and this is the constant term minus three. So this is the second factor of the polynomial. So it means the party nominee original polynomial Campbell it and has equals. I'm not hurting that again? Cambior Tennis six X x squared plus X minus 30. Marty blurred by X minus one. And this is the same result which we got in the previous example because we did the same example using the synthetic division instead of deviant. So this is car scented division, so we will do. One would exempt. So in this example, there will be a missing Terms are just put in the title that this is a missing term so you can refer back if you need or to handle missing term. So we have a question that fit Rise X cubed minus food exit square minus 25. We're X minus. Fight is a factor. This is given that X minus five is a factor. So no using the synthetic division. What we do here is we just right the coefficients off 1st 40 normal year, which is one minus four. And you know that there is no Ekstrom here. We have x cubed. We have extra square, but no extra. So we came it Ignore this fact when, when doing synthetic division if some term is missing missing time means that this term has a degree of three this time of the degree of two. So there must be a Diggory off one because this is the definition of a polynomial. But we don't have one is it is okay, but we have to take account for that. So how we do that? People Zero for that missing time here and then report there minus 25 off the constant. So the term is missing. We put zero. So only this is the difference. Otherwise everything is same. So this is X minus five. So we should put flame here. So what we do, we bring first number state of it on. And then we multiplied this five with one and report here. So five into one is five. Then we aired. These two numbers don't subject, so it will be one here. Then again, we multiplied this five with this one and put it dear. So we put five year and we had five and zero. We get fight again, do the same thing. We put this five year multiply it is five with five and put it there. 25. No beard, these two, and we get zero so that the zero confirms that we have done the right thing. We have done that correct procedure because if this is effective and we should get the remainder so no over is retained. Polynomial is a four degree. It is a degree too, because we're the wording our degree to report a normal with their degree One party nominee . So what is certain is excess choir. So this is the coefficient of X Esquire. So X is quite have the cover. You know, one This is the extra So plus X And this is the constant plus fight. So this is the second factor of this party nominee. So this party normal can Billiton is affected off this and X minus fight. So this is very simple. This was just one a little tip that if some term is missing So how you handle doctor so no , you know, many metres off fact arising Pollina meals, you can take common. You can use the estimated you can use any of the mathematical identity. A plus B, only SKorea manners be only square are a score minus B squared of their two terms. If there four times you can do grouping off two times together than the grouping of two terms together, and then you find the fine defectors. So, no, you know many methods, effect or ization. So I am confident that you will be able to fit. Tries any type of party nominal 234 Whatever terms out, and whatever the situation, you will find out where to factor is it if there are perfectly square first, look for the perfectly square. If there's a perfectly square look for that, if there is a mathematical identity, look for that. If not, then you the easy matter. If there four times just grouped together and other two together, try to find out what is common. If one factor is given, we just do their division of synthetic division and find the second time, and then, on the second time, try to fact tries it again. Whether you are able to fill tries, it are not so in this way. For example, if we are able to fact tries this trip so we can for the undertaking this, I'm just saying that if we can able to use any of the method like a C matador, the any of the identity matter, So if we can do there, we will do that. And then we will have three factors. If not, just leave it there. So know your you know. You know, many ways to factories a polynomial. It is a very important skill. And we will be using this in solving algebraic equation, especially the quadratic equations, because these were just the expression. These were the polynomial expressions. These were not the equations, but we can use the same skill to solve equations. 40. ALG 35 Types Of Algebraic Equations C: Types of algebraic equations. If an equation has this form, 3x plus two equals 0, there is only one type of variable. One variable there is no y, then this is a single variable equation. This is car, single variable. So just like polynomials, we are classifying the equations as based on the number of variables are the degree of the equation. So this is single-variable equation based on the, now what are variable? There's only one variable. Now. This classification is based on number of variables. Single-variable are two variable equation. No derivative classification which is based on degree. Based on degree. The largest exponent of any term is the degree. So we have an equation, phi x is square plus two x, y minus five equals 0 to what is the degree of this equation? So this is the first term, just like it is a polynomial as well. Normally corrosion. So you know how to find the degree of the polynomial and you add the exponents. So exponent of this is to disturb the exponent of this term is one, that exponent of x is one, y is one, total exponent is two. An exponent of this term is 0. X is 0. Here y is 0 because anything raised to the power 0 is one. So the maximum exponent is two. So the degree of the polynomial here is two. The degree of the algebraic equation is two. So this is called a quadratic equation. Quadratic equation because the degrees too. So I should have taken first the linear example. So here the degree is one. So this is car linear equation. If the maximum exponent is one, then the equation is linear equation. So this classification is based on the degree. Let me take another example. And we tried to define what type of equation is this? 5x minus seven equals 0. So this is a single-variable equation. Single-variable. And also linear because the maximum exponent is one. So it is a single variable linear equation. So what about this x squared minus five x y plus phi? So it is a two-variable quadratic equation where evil are multivariable. But because here two variable quadratic. Because if the power is two, we have a name for the quadratic equation. And we also have a name for the maximum exponent is three. This should be equal 0 because we are talking about equations start polynomials. So x q minus one equals 0. This is a cubic equation. So two-week equations are those equations in which the maximum exponent is three. But these are carved cubic equations. So you know now what our linear equations in which the maximum exponent is one. So there is only 3x 3y. So this 3y minus five equals 0 is also a linear equation in y. If you want to write it complete. So it's a single-variable linear equation in y. So this is a single variable linear equation in y. So this is a two-variable quadratic equation in x and y. So this is the way of classifying the algebraic equations. 41. ALG 37 Two Basic Methods Of Solving Equations C: Now the next thing I want to highlight is methods of solving equations. Equation has an equal sign and expressions on both side. So what I am telling here that these matters are the very basic set of rules which we will be using to solve any linear equations are quadratic equations at any form of one or two equations in one variable, two variable. Whatever equations are, there are some specific methods for, for example, for quantity equations, for linear equations, there are a lot of methods. But what we are talking here is not any method, but all those methods will be using the principle of these two matters. So what are these two methods? These two methods are the two ways of solving any algebraic equations. So the first is card balance method, and the second is card reward sing method. So this method is based on a logical statement that equals add E2 equals r equal are still equally. So you have $10.10 dollars. We bought are equal. Both are equal dollars. If somebody gives you $5 and give me $5 equals edit to equals, equals D2 equals, so we will still be equal. So this is the basic of this balance method. So what it says is that when we are solving, as I mentioned earlier, that solving algebraic equation is not a random process. It's not something you have to memorize. No. These are very well organized procedures with the reasoning. So this is the reasoning that if we have an equation, there is some X, some terms here, and some terms here. If I do the same operation on both sides, if I add something here, and if I add the same thing here, if I multiply something here and multiply the same on the other side, if I divide this on this side and divide the same on the other side, I am not changing the value of the equation. I am just trying to get the solution, trying to simplify whatever I think that is best in this situation. So I am trying to simplify. I'm not doing anything wrong. So this method balance method says that you can do the same operation on both sides of the equals sign. If you think that this will help you to get the value of x are any variable. So this is the balance method. The other method is reversing matter. When you are very comfortable with this balance method, then you can do that. If we're seeing better. We're seeing method is same as balanced method. But we skip one step. We just skipped one step. And when I will be explaining these matters, then I will explain how it differs from the balance method. It doesn't see anything different. It doesn't say that we should not add equal operation, no. But it tells us another way of skipping one misstep and making the solution process a little shorter. So this is where traversing method is and the reversing method, one thing you should always keep in mind that when using reversing method, you always do one step at a time. This I will explain further when we'll then this matters. So these are the two basic matters. Balancing method are the reversing matter. In the beginning we will be doing balancing method. And then after a few exercises, we will also do reversing matter. And when you become more experienced in algebra, you normally use reversing method. This was the first introductory lecture about solving the algebraic equations. We have not solved any equation. We just described few of the essential thing. Now we will be using one of these methods to solve linear equations. 42. 01 Method of Solving Equations with Grouping Symbols: So in this lesson we will be sore doing algebraic equations with grouping symbols, for example eight minus three one plus three X equals minus four. You can solve this equation either by using balance. Matter are by using very worsen matters. So there are four essential steps but any of these matters. So the first step is toe open these brackets to open the breakage and then what you do you take all the terms having the variable to the left hand side and all numbers put the right inside of the region. So first you open the break it, so you will get it. Minus three team are deployed by one is three and three multiplied by the X minus three more right by plus three X is minus nine X equals minus four. So you keep disturb on the left hand side because it has x so minus nine x. And in order to remove this eight, this is plus eight. So you weird minus eight. Using the bennis matter, I have already explained the balance method so minus tree. So you weird plus three. So you have done putting extra year blustery and minus eight. So we will do these things on the right hand side to balance the equation. So minus four minus eight plus three their terms, which we heard it on the left hand side with outside on the right hand side. So this real lead us to minus nine. X equals these towns Will be cancer minus four and minus. Later. Minus 12 plus three is minus nine. So the next step is to isolate the variable on the left or insight. So we have done the first step. So we didn't need Foster step in this equation because there was no very ever term. But we have done the system we have. We have eliminated all the numbers from the left hand side, so eliminate all the numbers from the left inside. So we have done this and no the final step. Isolate X. So in order to actually takes, just you have to rewire this by minus nine. So therefore we have to wear the same on the inside and this will be cancel. Oh, and X equals one. So this waas, if we're using the bennis matter, know if we're using the reversing matter. So steps are essentially the same. This is preferable that you perform only one step at a time. But in reversing method, we don't do this balancing thing. What we do is we take these things to the other and other side of the equation by changing this with their additive inverse and it gets even worse off. It would be minus eight every day. Even worse of ministry would be three. So we really remove from this side and we will additive and work on the other inside. So this is what we do in reversing method. So the steps that essentially seem use that this table to property to eliminate the Prentice's this adjustable to property opened the records Same step as we do this. The streaming step is being done here. No, Instead of eliminating, we're taking the terms. All the variables would you write in sight? So we take our terms having variables from left hand side to write insight and all variables. So we take all numbers from left hand side to write insight because we need our end dessert . It was something like this. X equals once our very able should be on the left hand side and number on the right and saying numbers the value off the variable. So this is what we want. So the same thing, Mr Devil immunity. We are no taking the numbers to the right hand side and taking the variables to the left inside. If, if there are somewhere labels on director inside were people doing an example in which variables will be on both side of the equations. But for this equation less good in using the reversing matter. So the steps will be like this. Eight minus three, minus nine neck were just opening the parentheses equals minus, for the first step is exactly the same. Or instead of balancing, we just take this air to the other and the side of the bracket. So we only keep minus nine x on this side. So minus nine, X equals minus four. When this year it goes on the other side, it becomes minus C on this minus. Did he goes on the other side? If you want toe booed off only one step of the time, you can leave it like this and looking minus three here. Then in the next step, Ministry minus nine X equals minus 12 the next step. You take this to the other end other side of the equation, and it will be plus three. So minus nine X equals minus 12 plus three. Reading was minus nine minus nine X equals nine and no university method. If something is being more deployed on the left hand side, we divided on the other end, other side of the Korean So x and we take this minus nine to the other side of the question and it will be divided. And X will be what? Because this will cancel. Oh, so you can use any of these meta very simple. Essentially, the steps are saying so. No, we will do quickly, few examples and without any further explanation of the matters that we just do for examples. 43. ALG 36 Infinite And Unique Solutions C: If we have an equation of the form x plus y equals ten. So what would we the solution? Can I find a solution for this equation? I don't like, I don't know why. Yes, I can guess, for example, if I say x equals two, Y equals to eight, then by putting these value x plus y, we get two plus eight, which is 1010 equals ten. So equation is satisfied. So it means this is a solution. So this is one of the solution. But what about if I put x equals three and y equals seven? I still get ten equals ten. So it means this is also a solution. So this equation has infinite solutions. So this equation has infinite solutions. On the other hand, if I have an equation of this form, x plus two equals 0. So what is the value of x which satisfy this equation? Minus two? If I put minus two here, minus two plus two is 0. So it means x equals minus two is the solution of this equation because it makes the equation true 0 equals 0. But this solution is a unique solution. It is a unique solution. It has only one solution is to equation. Has only one solution. Only x equals to satisfy this equation. So in most of the cases in algebra, we are trying to find the unique solution. So this will be our goal to find a unique solution for the equation. No, Hakeem, I find a unique solution. What is the requirement for the unique solution of this type of equation, which is that two variable linear equation. These are two variables. So for b type of equation, if there are two variables, we need two equations. We must have two equations to solve to get a unique solution for these type of equations. If, say we have one equation, x plus y equals ten. And we have another equation which is x minus y equals two. Nor we can find a unique solution. Nor we have two equations and two unknowns. Two equations. And two unknowns are two variables. So when the number of equations and number of unknowns are same, then we should be able to find a unique solution. Otherwise there will be infinite solution if we have only one equation and two variables. So this you should always keep in mind that if you have two variables. And remember that if a, b, c are ME constraints argued, it should be explicitly mentioned in the question. For example, if we have an equation, x plus y equals ten, and Bx minus y equals two. Just like this equation, just like this. But now we have eNB with x. It looks like that a and B are variable. A and b might be constraints. Whatever the value of a0 and b0 is, is, it should be explicitly mentioned in the question. So if you find that a and B are constants and they have certain values, then you can solve this equation because there are two variables, two unknown. But if a and b are also variables, then this equation cannot be solved, cannot get a unique solution. When we, when we will be solving equations, some WIC or this equation cannot be solved. So it means that it doesn't have a unique solution. So if there are some constants, so those should be explicitly mentioned there, there, these are the constants. And then you should know that read up, this is a solvable set of equations for these two is also called a set of equations. Set of equations. So whether the set of equation can be solved or not. So let's take this example. Now we have a unique solution for this. I am not solving equation. There is a set procedure to reach that solution. But at this time I'm just telling you that I, because I made these questions. So I know that if x equal to six and y equals four, x equals six, and y equals four. In any of these x, in both of gays or any of these equations, these will be satisfied. X plus y is ten, and x minus y is two. X minus y is this too. So it means now we have a unique solution. We have two equations and two unknowns. So the number of equations must be equal to number of unknowns to get our unique solution. I hope this makes sense. 44. 02 Examples of Equations with Grouping Symbols: Okay, so, no, we will do few examples. So when we have already done so, the second example is five multiplied by four X minus two equals minus 70. So far step open the briquettes. Five months great by forties, twenties or 20 x five to minus two is minus 10. It was minus seven. So take this time to the other and other size. We're solving this using the reversing matter. So 20 X equals minus 70. And when we take this Afghanistan to the other state, it will be blessed. Them for 20 x equals minus 70 plus 10 is minus 60. No, we have to. I select we're using this reversing method. This is Toby seem in both matters isolated the variable, so twenties with the variable. So we take this 22 the other sites which is being multiplied. So it will do rd when it goes to the other side. So now you can simplify 2120 threes of minus 30 X equals minus. Today is your onset, so you can check it to substitute this value into the original equation in the left hand side and solve it. So fight multiplied by four into minus three inside the bracket minus two. So the ultra record closes here equals you don't have to write equal because you just solve these and it should become minus 70. So far, you four more to president. Ministry is minus trail and minus two. So five marty paper minus 12 is minus 60 five. Multiplied by minus two is my understand, And this becomes minus 70 which is equal to the origin of value on the right in sex. So we solved the equation on the left hand side, and we did it at the sea will, which was on the right inside. So it means our answer is correct. So over tarting them police six multiplied by one plus X equal to 12 multiplied by X minus to d. So we will do. Two examples were there are variables on both sides of the equation. So here, exes on both side of the equation so far, step stayed the same. Open the records. Six multiplied by 16 multiplied by x. So six plus six x equals drive more depraved. Way exceeds 12 X. Greg Martin Prado, minus three is minus 36. No, you have toe. Bring this term on the left hand side and you take the numbers to the right inside. So you do. You can do this simultaneously because terms are being added to. There is no confusion, but it's better to perform one step at a time. So we take just 12 x on the left hand side. We keep the other sometimes seem so. Six plus six sex minus 12 x c plus 12 Brexit goes the stern minus 12 X equals minus 36. So now this is the note, and they're like terms. You have to combine those like terms at every step. We have to check the like terms. So these tour like like times are the times which have the same base with the same exponents exit the base, and one is the exponent in both terms. So they're like terms. That means we can add their coefficients and pour the base besides so minus trailer and six when added. So it will do minus six, and we just pour the based because it is the same in water. There's where they're like terms so minus 66 No, this six, particularly the stakes two equals minus 36. And when this six goes to the other side, it will be minus six. So minus 66 equals both ar minus. So just aired them and put minus sign. So minus 42. So we have to isolate No, because we're using reversing matter. So isolated, no so minus six is being repaired with X. So we divide this and take this to the other state of the equation. Divided by minus six minus 20. What about minus is plus and 42 divide over six. It goes seven times. So seven X equal seven is the answer. So in this equation, there were variable on both sides. So when we check, we have to put the value of X on both side of the equation. And if the answer comes same on both sides, it means our answer is correct. So we're checking six one plus X is seven equals Well, exes seven minus three six one plus seven is eight. Well, seven minutes trees four. So 68 is 48 and therefore that is also 40 years. So it means left inside equal side Writer insight and ever answer waas correct. So the last example. We have any quit in two weeks, plus six one plus X. He cost well X minus to B plus 14. So we're using again. The reversing my turn first step is open the records, so Wicks plus six months Broadway one is six six months apart by Access XX equals 12 multiplied by X is 12 x and 12 multiplied by minus three is minus 36. That's 40. So when you see like terms, you always try to simplify them. So we have, like times two eggs and 66 on this side and these air two numbers, so numbers can be simply fired at each step. So two X plus six eggs is your dicks. No. Six equals 12. It's minus 36 plus 14 to a different number. Subutex. Smaller from the larger and put the sign of the larger minus 22. So there is a very able on the right hand side, and we have toe take all terms, having variables from the right inside toe the left hand side. So we have to bring. This number is variable, the lifter insert, so it takes minus 12. It's plus six equals minus 22. One step at a time. So therefore, I'm not taking this six on their site. So your text ministry, which is minus four X like terms basis, same. You just subject eight from 12 for the send off the larger number. So minus forex. No, You're taking these plastics on the other side of the trillions of minus 22 minus six. So minus four dicks. It was two number with the same sign, just at the number for the same minus 20 years. No, we have to isolate. I still latex. So minus four X is being multiplied divided on the other end site. So minus 20 years, you are two by minus four. So now you simply five minus two or minus is always plus, just like mine is more depraved. Reminisces plus miles. Divide about minus is plus or go seven times. So what? Exes? Seven. This is your answer. And we can check this since we have variable on both side of die quickly, you know? So we have to put the value effects in both sides of the equation. Support seven on left hand side here. Go into seven plus six one plus seven equals Great seven minus three plus 14. So do into 7 40 plus seven. Plus one is +88 months. Play for sixties for pH. Several ministries four. Former depraved but 12 is 48. Bless 14. So on both sides we get 62 equals 62. So it means the question is balanced so over. Answer waas Correct. So these were the four examples off equation with their 45. 01 Introduction to the Language of Graph: of the next topic is ordered piers and graphs. In order to understand ordered pairs and graphs, I will start from number line because you already familiar with number lines. What was a number line? We had a line hours underlying on number zero was in the middle and on the right hand side we had indigent numbers 123 and on the left hand side, negative into your numbers and import direction. The number line goes toe infinite. While we were using this number line, we were using this number line to represent numbers because, for example, Number one was represented with a line of land from 0 to 1 similarly to what represented from horizontal light from 0 to 2. And we were also performing some operations using this number line. So basic purpose off this number line waas To show the relationship or to show the graphical are visual representation off numbers. No, the purpose of graph is also seen. We want to represent points and lines, the visual representation off the relationship between numbers. Whole numbers are related to each other. What is there with your representation? So this is the purpose of the graph so, no, instead of one number line as you that we have to number. Line one is the horizontal number line and the other is vertical number like and there are two number lines. They constitute a plane. They make a plea on this plane. If we want to represent any point, for example, this is a point on this plane are this is a line on this plane? What we need. So how can we represent this point on this plane? This point cannot be represented by only one number. No, because there are two lines and they're acting together. So there should be some very to represent this point so that we can take into account both lines. So how we do this, we do this by taking the distance of this point from the horizontal and vertical lines. And then we say, Okay, we want to represent a point. For example, I take this point if I want to represent this point on this graph, so I will need to numbers now. So let us say that those two numbers are X and y, So I would represent this point with two numbers x and white. So What is the relationship between these x and y and how I will get these tempers. So the way I will get this number is that I will take the distance of the point from the word he collects is so this is the word tick Alexis to article Exes are card. Why exes? Why excesses positive in this direction and negative in this direction. This is why bar negative y axis. So the polity part of this number line positive xx is in this direction starting from origin. This is positive and this is negative X sexist. So no, we have to number light, so we call them Access XX is on the horizontal axis and the other is why exit, which is the word he collects, is so the question waas hope to represent this x way on this plane on this plane which is also called Cartesian Plane. Because it showed the relationship of the members know how to represent this number. What is this? X X is the horizontal distance of this number from the word he collects is so X is their distance off the number from the world. Take Alexis. Then why we call it X because it is in the direction effects. It is in the direction of positive X. So therefore it's positive X But this is the distance from the work tick, Alexis. Similarly, the distance of this point from the horizontal axis is why these distances, Why there could be in for a night point on this plane on this card is in plain. So in order to represent any point no, we need to numbers. And these two numbers work together just like these two lines. No, these two lines, they are working together. Before, we were only using one number line. That was very easy. But no. These two lines that working together this point represented by X y and these X ray also worked together and not only together, but they should be in the same order this beer off points because this is a peer of points . The spirit of point is card ordered beer because it has a order X is always first. Then why so what does it mean? Is that for any point on this craft? No. We need to do numbers. Not one number, but two numbers. You have learned what our access you have learned what is a Cartesian plane? What is an order? Pit And all this system is his guard ordinate system Because they are all acting together These two points directing together x and white So therefore these points are card org needs They represent a single point in space This is a plane can call it a two dimensional space as well. For the space these x and Y are dimensions And this plane is also called not only the Cartesian plane but you can also call it X Y plane. So these are the few terms which you should learn first. Then we can discuss Hotaru draw lines hard to draw graphs off other functions not only state lies, We will be drying graphs off lines, similarly graphs of functions and we will be discussing functions after this lecture This luxuries just to introduced to you What is graph water ordered Piers Hovey represent points on a Cartesian plane. What is the Cartesian plane and all these type of basic things? So no, you understand that in order to represent any point on the X Y plane, we need to numbers. Those numbers are in the form off a pier which is card order here. The first number indicates the distance along The positive XX is the second number represent the distance along the positive. Why exit? So you should understand all these things, so you will be able to drop points. So then you will be able to draw lines. So first people be drawing points, Horta, drop points and hotel draw lines. Then we will learn how to draw lines and then we will learn hoe to draw the graphs off functions, function graphs, so this will be in the next lectures. 46. 02 Derivation of Slope Intercept Form of the Equation of Line: So in this lesson we will be deriving the equation of state like so, driving the equation of state line why is equal to him? Explicit is very easy. It's not a difficult job. You just use the concept off slow and you drive the question. So here it is over line. And this line has a point B. This can be anywhere on the light. Remember, whenever you are driving any generally question any generally question off a lime are any other function whenever you are driving any equation? So you take that X and Y rotation because this is a genuine imitation. So it means that this P can be anywhere on the line. So the second point we have is the point where this line intersects the Y X is and this important because this seemed like William is the Y intercept. And what is why intercept y intercept is the white coordinates off the point where the line intersects the Y axis are you can see that this is the distance on the along the Y axis at with the lining to six. So this distance is C. The distance from the XX is at this point where the light into six the y axis. So this is see in this equation and this P has coordinates X and y. So it means that X is the distance off Be in this direction. This is just a coincidence that this looks like to but he can be anywhere. It can be here, so this trend will change. So this distances X and the distance of P from the XX is is why so? This is why so P has, except by coordinators and the line. We're but the point. We're lining to six. The Y axis has a coordinative zero and C because there is no distance on X axis. So we have to drive the question of the light. Very simple. We will only use the concept off slope off the light slope off the line is all is a constant thing, is a constant. It never changes any particular light. The slope is always saying so. You can take the slope anywhere on the light if you take the slope here between these two points. But if you take the slow between these two points and if you take this low between these two plant. The slope is seen slow bodies. A slope slope means if you are taking slow between these two points, any two points So what you are checking you are just taking the changing their white coordinates the wire did by changing dead Exc ward in years, the issue off changing white with the change in X. Sometimes people say it's it's just the run and the rise descend that I don't like their definition. This is a run and this is a rise or whatever non technical type of definitions. The definition which I like their slope is the change in y coordinates. Do our did Wait Jinyan X coordinators, Any point, any two point on the line you can take and you can take the difference in their white coordinators. These air the white coordinators which fee right later with disordered appears to second number is always the y coordinate. So if you are taking slow between P and S, you just take their difference of Y minus C and X minus zero, and you will get the slow very simple. And in fact, we will derive this equation using only the concept of slow. So if you want to just see that What is the difference in white cord and actually me. And this is why. And this is seen So the difference in while coordinates will be This is dealt away. Why is the whole thing and sees up to this point Delta y equals? Why Mina Si similarly, what is their tax? This distance is X X is X coordinate of the point from the distance from the way a two this is X and S is at zero. So it means that Delta X is s minus X minus zero. So delta X equals X minus zero. Are you considered his ex? So what we will be doing this is only we're taking the slope between these two miles. But using these two points because slope is always saying you can take slow between any two points. So we will be saying that, OK, if he's any point anywhere on the line, people be here here. It's a general notation so any point. But we know that this point has our intercept off. See, we have denoted this we'd see. So we are taking the slow between any point and this point so so we say here, there's Lexie, that the slope of the line is m which is equal to change. And why do? Worded. What union? Next? Since we can take any book wines, any blue points therefore, we dick we decided to take slow between you take be and s two years ago. You can take every point, their work ward in that difference and take their exporting and different. You will get the slope. So no, you say they're m equals built away. Which is why I minus C and Delta X which is actually X no. You just multiply on both sides with X in order to eliminate this Eckstrom there denominator. And we get a mix on the right hand side and on the left hand side This X will be cares allowed with the sex and we will be left with Why am I gonna see no bring see on this side ? So M x plus e equals Why are you can switch the position due to community property if a equals B because it and we have drived a question off the state line sourcing. But just take the slow between any point p and the point where the line intersects the Y X is and you will get the slope. Intercept. Form off the question off line. This is called slope intercept form. So this is the slope intercept. Form off the question of line. And this is the most common formally used form off the question of life. 47. 03 Properties of Straight Lines: So in this lesson, we will be in trying to understand the state lines. Whole state lines behave what are their properties and hope we can understand state lines in a simple and efficient manner. So, first of all, I would like to mention that this lecture is for your reference. Whenever you have any problems in understanding the state lines are solving problems off state lines. You can come back and you can have a look at this lecture. So what is the line? Whenever we connect two points, we make a line. These points can be very close to each other. These points can be very far from each other. And this line is that in finite light, so at this point can be very close, very tiny distance between them. Even then, this line will be in for a night because when we said line, so line means and in for night state lines, which goes in the direction in which we defined it. So, for example, this line I defined using two points P one is minus 14 minus one on the X axis is why you along the way for allowed the Y axis for this And the other point I took for this line, WASPy to which is four minus 64 units along the X axis and six units in the negative Y direction. Negative six. So I just joined these two points and then this line is going in front of distance. Similarly, I joined these two points and this line is going in for distance. So no, If you look at the slope of this line so how you get the slope off Any line at how you draw a line if the slope is given So this is the first thing we try to understand. So So in order to get the slow what you can do is you start from any point so you can go right. You can go left If you go left, you go. Palito xx is this is excess and this is pallor to XXI. You go towards left and there you go downward and you reached at the line again and you measure the distance in the extraction and in the Y direction. There you take the ratio off changing. Why do everybody x and you get the slope? In the previous lecture we already drive the question of this state line. So this is a 1,000,000 of this line y equals X plus two. If you compare this equation with the equation off the line AmExp Lasi, you will find that M is one here and is one. And you can see that this is one unit in the Y direction Do added by one units in the extraction, you will get one. So this is the equation of the state line. No, this is the point. Where this line intersex. Why? Exes. So this is the distance from zero to this point, which is two units one and two two. Unit is the intercept. Why intercept off this line, which is highlighted here? Why? Intercept off this line is two units. This is why Intercept? No, If you look at another line, this one, not this line. What? What is the slope of this line? This land has an equation. Why is equal to minus to express to for example? You don't know this equation. And you were asked toe derived a question of this line. You were given only poop warrants P one and p two. So hopefully you released this equation. Just find the slope of this line and the point where the line intersects the Y axis. So the line intersects the Y X is there too. So you will proceed like this. You have p one. For example. You were given only P one minus 14 and you were given Peter. Somebody asked you that. These other two points which makes the line you connect these two point to make a line that they say that these two points lie on the line find the slope of the length. So what will be the salute? It's love will be equal to difference in y coordinates. So you can take the difference in both ways. Like tell the Y divided by death tax. So what you do if you are taking why to this is this is, for example, this is 0.1. So we say this is X one and this is why one and point do we say this is extrude and this is why No. Remember this Delta y you can take like this Why? To minus y one. Do what I did by extra, all minus x one. And if you take the minus common from this, my Miss Coleman. Then this will become minus y two because the sign will change. And this will become plus way one. And you take the minus comin from the denominator. I just wanted to want to show you something. Therefore, I'm doing this otherwise you can state of a calculated, but I want to show something. If you take minus common, you will be left with minus x two. Bless X one. No, If you simplify this minus and this minus both minus one divided, it will become plus, just like multiplication when minus vertebrate with minus is plus minus two, ordered by minus is also plus. So then you will be left by. No, you can change the position, but I first write the same way minus y two. Bless way one Do I did by minus extra all Bless X one. No, you can switch the position because Putin arms are adding a plus B or B plus a seam so we can say y one minus y two and x one minus extra. No, look at this equation and this equation. This is the source off error because student do hear a lot of haters. If you big though, why to minus Webvan, then you must take extra minus X one you take. First point is B one and it's coordinate as X one y one and take the second point. Be doing his coordinators extra weight. No for delta y either you take while to minus one man extra minus X work are you Take Why one minus y two. But you have today X one minus X to hear. Otherwise the sign will change if you take once where two minus y one and divide it by expert minus 62 The signed with the sign of the slope will change. So instead off line going in this direction the minus Sloman it will be going in this direction is I'm going to explain next. But so just keep this in mind. So no, we decided this that we will use this formula while you two minus wife. So then this is over way too. So minus six minus minus way. What is four? You are did by extra, which is four minus this minus one. So minus six and minus For his minus 10 four, this will become +14 plus one is five, nor tend word by five is two and minus divided by pluses minus minus two is the slope. So no, you can see here that minus two is the M So know Valli can have either A positive slope of the line can have a negative slope So what type off line will have a positive slow? So if you look at these if the line is going like this that used whatever may be the y intercept if even this one are the other line could be this one Or it could change like this one. In all of these situations, when the line is going towards the right and left like this upwards and on the leftward direction you can see the slope will always be positive because from any point they're safe for this light. If I go negative extradition and one negative X direction and negative y direction so negative, why divided by negative X will meet this positive. So if I go positive, then I have to go positive in order to reach the line. So it means positive, divided by positive will make this low positive, nor the lines which are going in this direction. What will happen Like this one here if I start from this point in order to find the slow and go this way So this is negative x negative. They may have to go positive in the way so positive way. Because Lopez of issue of the change supported T Y and negative X, the slope will be negative and even if to go on the other direction, for example here you can do like this. If, for example, you go why you bought it do like this and then you have to go Xnegative So seem so this line. Even if you go in this coordinate what you were doing, you go first Xnegative then why posit to so In any case, if the line is going in this direction even a little bit off angle like this, even a little bit of angle like this does love will be negative. And if the line is going towards like this upward direction, then the slope will be positive. So these are a few things. What does the question off line means when we say y equals X plus two? We know that low bar in to see what other information we can get from here. So where does it mean is that for example, I have X and I have few values affects one, 234 And there's the only four values I drive for. Well, so what will be the values of white? So if I put X one when X equals one, y equals was one plus two three. So it means this 0.1 and three will be on the line. And if you look here, you will see one in there s direction and 12 and three this point actually lives on the line. So for every value off X, you can find the valley off y coordinate on this line. So this is what they because this is sometimes called the characteristic education off the line corrected 60 characteristic equation of the light. Because if you know the really off X, you can find the value of oil anywhere on the line. XX is also line. This is a line. So what will be the slope off X axis? So XX is has a zero salute. XX is has a zero slow because there is no change in white, a slope is defined as because this line If I take any true point on this line, for example, I take this point at this point. So what is Delta y Between this point, there's the 10 between these two points and what is their tax duplexes? One. So what will be the slope affects sex is delta. Why do or did by death tax, which is zero Doherty by one, which is zero No come to y axis. What will be the slope off? Why exes? Why X is also a line. Why Excesses airline deserted by excuse the light. So what did the slope of why exes. So what? This look Slow peas did the way between any two points to worry about Del tax between any two points changing y coordinate between any two points. Let's say I take this point and I take this point. So what is the vie coordinate of this point six. So I say six minus. What is the y? Coordinate of this 60.3. Okay, A guard. A genuine way. What is the exporting it of this point zero? Because this is on my exes. So zero minus. What is the coordinator off this point? What is the X coordinate of this 0.0.0. So it means the slope will be three worded by zero. And what is if you do at the number of is zero What? You get in for a night. So the slope of why exes is in for a night The slope of the way. A crazy in for a night. So what is the practice stick equation of y X is what is the equation which describes why you exist So we say they're cracked. A sticky question of why exes is X equals zero on y X is X equals zero. This is like Well, you know the way, so you can see that all way. X is X zero. Similarly, you can see the queen of the X axis. There's y equals zero. This is a question of the X axis or all XXI. Why zero? So this is characteristic equation of these excess. For example, take this line. There is a line which is parallel to work. Sex is this life. So what will with the question of this line on all this line? Because this is This is a line paler to accept it. So you would think OK, the exceptions other equation y equals zero. So this land should also have the same type off equation. Yes, it is because it has. Why equals one all this line, what is equal to one? So I question of the land media question must be true for every point of the line. So only this week was one can be true. For every point of this line, no other equation can be true for this light. Similarly, the line is like this. If the line is pelant, why exists if the line is Ballard one axis So it must have any question something like this . X equals 123 but minus three X equals minus trees a line which is like this which is Ballard worked. So if you see a line equation if you normal equation of the line is this. But if you see the question of the line is given as X minus three, so you can also say OK, this is an equation of the line. But this line is Ballard y X is because there is no way in this. So but these are special. Special equation off the lines. You can say these are the special. Normally, whenever you talk about straight line, you say state line has any prison off Wyche was MX plus C. So this is the normal equation of the state line. I'm just trying to enhance your knowledge a little bit so that you can better understand straight lines. Their state lines can be off any type. So if a state land is in this direction, even thought the slope will be because it can, there can be only two direction. One is this one. The other is this one. Otherwise, if the landing state then you're speller toe XX is. And otherwise if the line is vertical, it is better. Why exist? So then it will have equity in something like this. So this is what you should understand that state lines host your lands. Behave? No. I will show you what our perp, a nickel lines and what are badly lines 48. 04 Parallel and Perpendicular Lines: So in this Listen, we will be discussing what is a parallel line and B, what is prophetically like. So if you look at this line so this line is better toe this line. So no, what is the practice? Stick off these type off lines. If two lines are parallel ho, can we judge this thing from looking at the equation of those lines? Is there any clue in the equation of the line that whether these two lines are parallel are not So this is what we are going to them here. So what is the slope of this line Petri before? So this has a slope of one. If you go one unit on the left and one unit on the towards On the downward side, it means we're moving in the Manus X direction and then we're going my minus one in the X and that minus while in the white so sign is important. So solo peas defined as the changing why do harder by changing X so changing Why will be negative and changing X will also be negative. So to negative values negative one which is actually one because we have C, we can see that this is one unit, the wider by minus one, which is equal to one. And we can be seen from this. This is M The coefficient of X is one so we can see that this is one. So no, Look at this line equation of this life. What did the slope of this line. So from any two point we can take from this point and we can reach here. So if you go one unit on the left and one unit on the downward direction so we can get the slope of this line and this is exactly the same. So it means two lines which are parallel. They must have the equal slope. Otherwise, they can't be parallel. And if they have equal slope, they must be paired the lines. So no, look at the question of this line. Y equals X plus two. So too, is the y intercept. And this equation is Y is equal to X minus three minus tree is the Why interested? So it means No, you have one equation. Why was X Plus two and the other equation X minus three. The slope is same. The only difference why intercept? So it means if this is going at X equals this if this has the Y. Intercept of two, this is the Y Intercept of three. They're parallel, so there should be. Can you think that they should be in the parallel Lame going from or didn't. So what do you think? What could be the question of this line? My drag is really bad, but this is a straight line. Assume that this is a straight line and this line is passing through the origin. And this line, this is a state light, and this line is passing through the origin. So what will be the question of this life? So the question of this line will be why equals X plus the Y intercept, which is zero. So it means y equals x. No, look at this equation. Any point on this equation because every point should satisfy this equation. So this is a key. Why don't you should always keep in mind? This will help to solving problems, and every point on the line should satisfy the question of the line off that line. If a point lies on a line, it should satisfy the question off the line and satisfy means that when I put X and Y coordinators off that point into that line, the question of the line should be proved. So this means satisfies. That is why I mean that it should make that question of the line drool when I put X and Y coordinates off that point in today Quillian of the line. So it means this is this This line has a zero intercept. So it means the question of the line is equation of the line with passes through the region is y equals X. So it means every point on this line has should have both X and y value of the coordinates . Seems so. What is this point? This is one one. What is this point? This is to index to in the way to do so all those points which are on y equals X, they have same X and Y coordinates. So therefore, if I put this here, if I put the X coordinated at this point in this equation X is one equals. Why is one if I put the coordinator at this point? X is too wise too. So all those points on this line will have the same X and y coordinates. So I understand that by looking at the question of the line Y equals X, you can say OK or this line is passing through a region and it passes like this, which means every point on the line should have the same X and y coordinates. It goes to infinity, but it still have the same court in it. This point will be on the light because it has four x four way. So this is the only light with passes through all these points which have seen Maxima coordinators. So this really know what the world and a deadline which bosses in the same manner but in the opposite direction. Whatever this line, if I die another line a little bit, Milburn. So what do you think? What we with a question of this life, what will you be? A question of dislike. So in order to find that even if I if you look at any point for example of this point, what is the X coordinate of this Point X is equal to minus one. What is the way? Coordinate off this mind. Why ik was one. Okay, there's moved to this point. What is the x coordinate of this point? X equals minus four. What is the Y? Coordinate of this point? Y equals four. And no, you can find the slope because I intercepted zero. So we know, for example, this is the equation of this life. Let me right in the bill. This is a question of this bill. You line em explore. See, I go poop wires on the line so I can find him. So what is M and weak was the difference of the white coordinators. So let us take this point has B one and dispute. So I'm taking this minus this. So four minus one four minus one. Divided by minus four minus minus one. So four minus one is three and minus four minus one will become plus one minus four plus one is minus tree so the slope will be minus one. So when slope is minus one, no, I can put in the equation. I can say why equals to love his minus one. The equation of this line the blue line will become Why was minus X plus zero? I don't have to write plus zero Why equals minor sex? So a question of the red line is y equals X and equation of the blue Line is why equals minor sex. And these two lines are perpendicular and look at their slopes. If I might be, play the slope off the red, for example. The slope of this line is M one, which is one and slow off the blue line. I call it m two, which is minus one. So if I multiply m one with em too the onset is minus one and this is the property of the perpendicular line. They're slope off the two lines which are perpendicular the product off. Their slow is always minus one. So this is the property of pumpernickel, like in parallel lines. So I can say for two parallel lines barely lies em one equals m toe. Their slopes must be seen. We have checked this already here. This line and this lime are parallel. Imagine that If you tell this XX is toe 45 degree, you reach a deer And this way exit was also detected by 45 degree so x excision way Exes are particular. So therefore this red line and blue line will also be perpendicular to each other, and the product of their slopes is minus one. And this is true for all political rights. The derivation of this I'm not going to do with this is not required, but this product off the slope off Cooper particular lines is always minus one, so the slope of two parallel line is always seem, and the product of the food product off the slopes of two lines, which are perpendicular, is arteries minus one. - So these are the two things you should always keep in mind when dealing, for example in question, it says, This is a line, and this is a point. Another line, which is parallel toe this line. So sometimes you get questions around these type off knowledge debt. Rather, you know, these things are not so. Sometimes you get questions based on this knowledge that you should know their two lives there. Parallel there, slope should be seen, and you will find the type of question in my quiz and in my homework, because we will be doing a lot of this. No one last thing I want to highlight in this lecture is so one last thing I want to highlight in this lecture is that we have soared equations, for example. These are two lines. We have the equation y equals minus two X plus two. The question of this line and the question of this line is y equals X plus two. So let us solve these equations. Let us told the SoHo this all simultaneously an equation are too linear. Equations. We just subject this question from this one. Just change assignment. Subject Why? Minus y zero and minus two x minus X is minus to be X and plus to manage two equals zero. So it means minus and minus zero. So it means minus three X equals zero. And if you divide by trees, toe at X equals zero. Okay, Now put X equals zero into any of these equations. For example, I put in this one the original equation. So why was X plus two? I put this here. So why IK was zero plus toe. Why equals two? So if we were solving these two linear equations, so what was our solution of a solution would be 0 to 0 X and are you in the white zero toe would be over solutions literally off executed. The value of I know. Look at the intersection of these two lights. These two lines are intersecting at zero and two. So what does it mean? It means that the solution off two lines if you were asked to find out the solution of colon using Graf who just dropped to allies and find out their point of intersection. This point of intersection is the solution of two lines. We descend. The solution of the equations of two lines are two lines, you can say because lines have linear equations. These equations are linear equations. Both lines heavily nearly Quincy next and white. So the solution of these two equations, if they were, we were solving them mathematically. So this is the mathematical solution mathematical solution. But you can also solve the equation to state line are too linear equations because every linear equation is a straight line and inclusion, so you can also solve the linear equations by using the graph. So graph is so useful is the quick matter to can quickly door the lines, find two points, draw the line and find where they intersect and find the quarters of that point very simple . So this line and this line intersecting at this point, order the coordinators of this world. There is zero X because schooling upward and who is the vice ex wife is the solution of these tool in your equations. So no, you know how to find the solution of two steered lines are stoolie near equations in two variables in X and y. If they questions are only next, then of course, you can describe it a special case off a line, but then finding a solution is different. Easiest one step solution are you can use the ballast matter what we were doing in the earlier chapters off this course. But if it's ah, it's if. But if they're two simultaneous linear equations in X and way, you can solve the equation using only the graph. Just draw the graph. Rather, two state lines find the point of intersection. That will be your solution. They see your ex, and this is your way 49. 05 Solution of Pair of Simultaneous Equations in x and y by Using Graph of Lines: we have soared equations, for example. These are two lines. We have the equation y equals minus two X plus two. The question of this line and the question of this line is y equals X plus two. So let us solve these equations. Let us all these so hopeless all simultaneously in equation are too linear. Equations. We just subject this question from this one. Just change the science subject. Why? Minus y zero and minus two x minus X is minus to be X and minus zero. So it means minus three X equals zero. And if you do it by trees, toe at X equals zero. Okay, Now put X equals zero into any of these equations. For example, I put in this one the original equation. So why was X plus two? I put this here. So why IK was zero plus toe. Why equals two? So if we were solving these two linear equations, so what was our solution of a solution? Would be 0 to 0 X and two in the white zero toe would be over solutions literally off, except to develop. Right now, look at the intersection of these two lights. These two lines are intersecting at zero and two. So what does it mean? It means that the solution off two lines if you were asked to find out the solution of Toolan using Graf who just dropped to lies and find out their point of intersection. This point of intersection is the solution of two lines. Week descend. The solution of the equations of two lines are two lines, you can say because lines have linear equations. These equations are linear equations. Both lines. Have we nearly Quincy next and why? So the solution off these two equations if they were river solving them mathematically. So this is the mathematical solution mathematical solution. But you can also solve the equation to state line are too linear equations because every linear equation is a straight line and inclusion. So you can also solve these linear equations by using the graph. So graph is so useful is the quick matter to can quickly door the lines, find two points, draw the line and find where they intersect and find the quarters of that point very simple . So this line and this line intersecting at this point, order the coordinators of this world there is zero X because it's going upward. And who is the wife? Ex wife is the solution of these tool in your equations. So no, you know how to find the solution of two steered lines are stoolie near equations. This is your ex and this is your wife.