LINEAR EQUATIONS IN TWO VARIABLES|CLASS 9| CHAPTER 4| NCERT| COMPLETE CHAPTER| INSIGHTFUL MATHS
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00:00Hi everyone. Welcome to Insightful Maths. In today's session, we are going to discuss
00:05linear equations in two variables, which is chapter number three of class 9 NCRT.
00:11So before going ahead, if you have not subscribed to my channel, please like and subscribe.
00:17So let's start. Let us understand what do we mean by a linear equation.
00:24This word linear means wherein the highest power of a variable is 1. For example,
00:31if I give you something like this, this is 2x square plus 3 equal to 0. We are talking about
00:43linear equation. Equation means there has to be an equality sign. First thing, it is an equation.
00:50Left hand side is there and right hand side is there. Linear means the highest power of the
00:56variable has to be 1. But we can see here only one variable x is there and what is the power?
01:03Power is 2. So it is not a linear equation. One variable or two variable is a separate thing.
01:10First of all, let us understand how to decide if this is linear or not linear. Here the highest
01:16power is 2. So it is not a linear equation. If I write it like 2x plus 3 equal to 6. Is it an
01:25equation? Yes, because there is an equal to sign. Is it linear? Answer is again yes, because there
01:32is nothing written here. If nothing is written, it is automatically 1. So yes, it is a linear
01:39equation. But how many variables we can see here? Variable means the alphabet. Other than the
01:44numbers, only x is there. So this is linear equation in one variable. If I write it like this,
01:532x plus 3y is equal to 9. Now it is a linear equation in two variables. Why linear? Because
02:04the highest power of x is 1. Highest power of another variable also is 1. Two variables are
02:11there and there is an equal to sign. All the criterias are satisfied. So this is linear
02:16equation in two variable. So when an equation has two variables, I have just told you both of degree
02:241. Degree means the highest power. That equation is known as the linear equation in two variables.
02:31We have just seen. Now let us learn what is the standard form of writing the linear equation in
02:38two variables. The standard form you can see here, the standard form of a linear equation
02:44looks something like this, which is ax plus by plus c equal to 0. Please understand right-hand
02:53side should always be taken as equal to 0 and all the things to be shifted to left-hand side,
03:00where a, b and c. All these things are real numbers. This is the symbol for belong to.
03:08This one is belong to, where a, b, c are real numbers and if it is a linear equation in two
03:16variables, a and b should not be equal to 0. If one of them is equal to 0, that will become
03:23a linear equation in one variable. Let's take an example. If I write something like this, 2x
03:31plus 6y equal to like. It is a linear equation in two variable, but it is not written in the
03:39standard form because in the standard form, x, y and all the variables, any other thing should
03:45be there on the left-hand side. The right-hand side should just be having equal to 0. So what
03:51I am supposed to do, take this line on the left-hand side and it will look something like
03:56this. 2x plus 6y minus 9 is equal to 0. Now this is the standard form of the linear equation in two
04:05variable. The number or the coefficient which is attached to x, that is your a. So what is your a
04:13here? This is your a. So you have made ax. This number or the coefficient attached to y, that is
04:22your b. It is by and the last number along with the sign, it is minus 9. So what is your c? It is
04:31your minus 9. So it is now converted to this form, wherein a is 2, b is 6 and c is minus 9.
04:41This is the way you convert any given equation in the two variable in the standard form.
04:48All the things to be taken to left-hand side, right-hand side equal to 0. If there is written
04:54nothing is given for y. For example, it is given 2x is equal to 7. We have to write this in the
05:02standard form, of course in two variable, but I can see there is nothing for y. If nothing is given,
05:09consider it yourself as 0. So 2 into x plus 0 into y. Now two variables, take the 7 to the
05:20left-hand side, it will become minus 7 equal to 0. Now this is the standard form of the linear
05:27equation in two variable. I hope this much is understood. Let us move ahead.
05:33Now the solutions of the linear equation in two variable, it is a coordinate or a pair of numbers
05:41which can satisfy the equation. Please understand since the linear equation in two variable has two
05:48variables. So the solution means a coordinate we are talking about, a coordinate xy such that
05:57such that the value of x and y, if we put it in the equation, it will satisfy the equation.
06:05Satisfy means the value of left-hand side and right-hand side will become equal. So if I say
06:11I have a linear equation, let us say 2x plus y is equal to 3 and I am being asked is 0, 0 a
06:21solution of this equation? Solution is a coordinate. It is a coordinate. It has something,
06:28the value for x and the value for y. First number always tells you the value of x,
06:35second number tells you the value of y. So if I put these values in the equation,
06:41if left-hand and right-hand side is equal, it is the solution, otherwise not. Now 2 into x means
06:482 into this 0, this will become 0 plus y is 0 here, 0, 0 plus 0 is what? It is 0. So that implies
06:570 is equal to 3 which is not equal, which is not possible. So 0, 0 is not the solution of this
07:06linear equation in two variable. This is the solution. Now please understand there are
07:13infinitely many solutions for a single linear equation in two variable. If you are asked how
07:21many solutions of a linear equation in two variable are possible, your answer will always
07:26be infinite. Why is that so? Because please understand if these two are coordinate axes,
07:37this is your x and this is your y, two coordinate axes are there and this is the Cartesian plane,
07:43linear equation in two variable will always be a straight line. It may or may not pass through
07:49the center, it is not the criteria. For example, that it is a line something like this. Now all
07:57the points which are lying on this line, please understand all the points which are lying on this
08:05line, it will have, if I talk about this point, it has some x value, it has some y value. In the
08:13similar manner, this point has some x value and some y value. All these points are what? These
08:20are the solution of this linear equation in two variable. Why solution? Because these points are
08:28lying on this line and line is what? Line is always infinite. It can be extended indefinitely
08:36in both the directions, so there will be infinite point and hence a linear equation in two variable
08:42has infinite solutions. This I have already told you, a pair which satisfies the equation is the
08:50solution. Graph of the linear equation is a straight line, I have already told you.
08:57Since there are two variables involved in the solution, solution will always be in this form,
09:03in the form of a ordered pair or coordinate. The pair which satisfies the equation is the solution.
09:12Now let us understand how to use a graph, the graphical representation and how to find the
09:21solution of this linear equation in two variable using this graph. I have used this graphical
09:28paper, this is x axis and this is your y axis. The easiest way or the method is,
09:35step number one, you have to make a table. You have to make a table like this,
09:42divide it in two parts, this one is your x variable and this one is your y variable.
09:49We will just input the values here and find the value of other one.
09:54Step number one, always the easiest method, first of all keep x 0. Keep x 0 in this equation,
10:02so that means 3 into 0, it becomes 0 plus 4y is equal to 12. This number has become 0,
10:12so 4y is equal to 12 means y is equal to 12 by 4. Are you getting the value of 3?
10:19So if x is 0, y is equal to 3. This is the first solution. See how easily we have got.
10:27Another easiest method like in the first one we have put h 0. Now what you are going to do,
10:34in the same equation now put y 0. If I take y as 0, are there many y 0 here? This portion is 0,
10:42so you are left with 3x equal to 12 at x equal to 12 by 3 which is equal to 4. We have got the value
10:51x equal to 4. Another one, I now put x equal to 2. I am putting x equal to 2. Let me put x equal
11:05to 2. Now have a look if x is equal to 2, 3 into 2 is 6, so my equation is now changed to, let me
11:14do your calculation for you here. If x is equal to 2, 3 into 2 is 6, so 6 plus 4y is 12. 6 will
11:24go to another side, so 4y is equal to 12 minus 6 which is 6 and y is equal to 6 by 4. 2 into 3 and
11:362 into 2 and 3 divided by 2 is 1.5. So what is the value of y you are getting? 1.5. Likewise,
11:44you can just put the different values and just 3-4 pairs are enough for you to draw a line.
11:52So now let me just mark the values here. This is the origin 0, this is 1, this value is 2,
12:02this value is 3. Please have a look, I need x till 4 maximum. So just 2-3 points ahead. 4,
12:105 and 6. First coordinate x is equal to 0 and y is equal to 3. Where is 3? This is 1, 2,
12:213 and 4. x is 0, y is 3. Where do we get x coordinate as 0? It is on the y-axis.
12:30The value of y should be 3. So we are talking about this point. y is 3 and x is 0. Now where
12:38do we get y coordinate 0? It is on x-axis. So take the value of x as 4, y coordinate 0.
12:46This is for us to check. You can either mark this point, hardly matters. x is 2 and y is 1.5.
12:551.5 lines somewhere in between. So it has to be somewhere here.
13:00Somewhere here, the third point. Now what you are going to do,
13:04join these points to get the line. We are here and we have to go till here. Please have a look.
13:13If I join this, now we have got this equation. This equation is plotted in the form of a line.
13:21You can make out it is a straight line. Since it is a line, make two arrows here and you can
13:27just give the value. What is this coordinate? x is 0, y is 3. Which is the middle coordinate?
13:34x is 2, y is 1.5 and which is the third one? x is 4, y is 0. There are several other infinite
13:46points on this line which all are the solutions. So how are we getting the value? First put x,
13:54get the value of y. Then put y as 0, get the value of x and some other different combinations of x
14:00and y. Just put the value of one variable and get for the other one. Fill this table,
14:06mark on the graph paper and that's your representation. I hope this much is understood.
14:12Let's move ahead. Okay, now it's a small chapter. We'll be able to finish it up in this video only.
14:19Some extra topics if you wish to see, please don't forget to see the next video. Okay,
14:24so question number one says the cost of a notebook is twice the cost of the pen.
14:31Twice means two times. You have to write a linear equation in two variables to represent this.
14:38Cost of the notebook is x and that of a pen is y. Okay, answer number one. Cost of
14:48notebook. It is given rupees x. Then cost of pen. It is given rupees y. Please don't forget
14:58to see whatever hints or the values are given to you in the question. Don't take it yourself.
15:04Cost of the notebook is that means x is is means is equal to. So x is equal to what is that equal
15:13to twice the cost of the pen. Twice means two times. So what is two times of y? It is 2y.
15:21We have bought this x. The cost of notebook is equal to twice the cost of the pen. We have bought
15:28this. Always ensure to write it in the standard form. How to write it in the standard form?
15:34Take everything on left hand side. So it will be 1 into x minus 2 into y and there is no c here
15:44plus 0 is equal to 0. This is the answer in the standard form. I hope this is understood.
15:52Okay, let's come to question number two. Express the following linear equation in
15:58the standard form and then you have to find the value of a, b and c. Okay,
16:05let us start doing from the first one. Step number one, writing it in the standard form.
16:12So question number two first part. Till now it is written 2x plus 3y is equal to 9.35 bar.
16:21Step number one, take everything to left hand side. So this is 2x plus 3y minus 9.35 bar.
16:32In the bracket, you can write the standard form of a linear equation in two variable looks
16:38something like this. h plus vy plus c equal to zero. Now comparing these two,
16:45this value is your a that means a is equal to 2. This coefficient attached with y it's your b
16:53along with the sign 3 and this complete value along with the plus or minus sign.
17:01So c is equal to minus 9.35 bar. Right? That is the answer for the first part.
17:10Okay, let us do the second part now. x minus y upon 5 minus 10 equal to zero.
17:19Everything is already there on the left hand side, you just need to find the value of a,
17:24b and c. The number attached with x coefficient nothing is there, take it one. So the value of a
17:31is one. Anything attached with y along with the side is your b. If nothing is mentioned,
17:39it is one and denominator is five. So which number is attached with y please have a look,
17:45which number is attached, it is minus one by five. Please don't forget to take the plus and
17:52minus sign also it is minus one by five. And this third number along with the sign
17:58minus 10 that is the value of c. I hope you are finding it easy enough.
18:05Okay, third part you can do easily. In fact, all the parts let me just quickly pick up the
18:11fourth part for you. Now, for the fourth part x is equal to 3y. Take everything on the left hand
18:20side, x minus 3y is equal to zero. Now it can be written x minus 3y plus zero because c is not
18:30there. What is the value of a? a is equal to one. What is the value of b? This one b is equal to
18:40minus three and what is the value of c that is zero. Okay, likewise, you can do every questions.
18:47Let me do just one more the last part. Five is equal to 2x and taking 2x on the left hand side.
18:56So it can be written like minus 2x plus five is a plus c is zero is equal to zero. Because
19:04five minus 2x or minus 2x plus five, it is one and the same thing. Okay, just to write it in
19:10the order I have written like this, this number a is equal to minus two, we are not having any
19:18variable y here. Okay, so you should have written zero y here. So minus 2x plus zero y this value
19:28b is equal to zero and what is your c the constant here that is equal to five in the standard form
19:35always ensure to write it first x then y then the constant so write it like this minus 2x plus
19:43zero y plus five equal to zero. Okay, this is all about exercise 4.1. Let's move ahead now
19:53for exercise 4.2. Which one of the following option is true and why? y is equal to 3x plus 5.
20:02It has a unique solution, two solutions or infinitely many solutions you need to tell.
20:10Is it a linear equation in two variable? Did I tell you linear equation in two variable always
20:16has infinite solutions? So answer is the third part. Okay, because it is a straight line and
20:22a straight line has infinite points. All the points on that line are the solution. That's
20:27your answer. Write four solutions for each of the following. Okay, four solutions you need to
20:34give you can give infinite it is asking just for four. First one is easy. I'm picking the second
20:41one for you. Question number two, second part, pi x plus y equal to nine. Okay, what did I say?
20:52First put x we have to find how many four solutions. This is x. This one is y.
21:00If I put x equal to zero, put x equal to zero. This part will become zero answer is y equal to
21:09nine. We have got one solution that is zero and nine x zero y nine. Now put y equal to zero.
21:18If I put y equal to zero, okay, so second part of zero, pi into x is nine. And what is the value
21:26of x it is nine by pi, you can wave it like this. So value of x is nine upon pi. Take it ahead
21:34because we have to find four solutions. Now, you have to put any value of x. For example,
21:41I'm taking x as one. So x is one x equal to one. So what does this equation becomes pi plus y
21:50equal to nine. So what is the value of y? It is nine minus pi, you can write here the solution
21:57nine minus pi. Likewise, if I put y equal to one, y equal to one. So this is converted to
22:07pi x plus one is equal to nine. So pi x is equal to nine minus one, which is eight,
22:15and x is equal to eight upon pi. So you have to write here eight upon pi. All these combinations,
22:23the coordinates are the solutions of this equation, different different values of x,
22:29you'll get the values of y. Easy enough. In the similar manner, you can do the first and the
22:34third part. Let's move to the third question. Which of the following are the solutions of the
22:41equation x minus two y equal to four and which are not? How do we check if the given value or
22:48the coordinate is a solution or not? You just need to put the values in the equation and check if
22:55left hand side and right hand side are equal. I'm going to do one part for you. Let me just pick
23:01this one. It is given in the root form. So I'm picking this one. Others you can easily make out
23:07x minus two y equal to four, put value of x as root two, root two minus twice of now y is given
23:18four root two, I'll write four root two. This is your left hand side. And this is your right hand
23:26side, which is given as four. Now this is root two minus four into two is eight. It is eight root
23:34two. If nothing is given outside one root two root two is already common. It never gets cancelled.
23:41When do it gets cancelled when it is dividing or multiplying? Okay, so one minus eight is minus
23:48seven. Left hand side is this right hand side is this which are not equal. So this one is not a
23:55solution. Okay, it is not a solution. Let me do the fifth part as well. Now put one one x minus
24:05two y equal to four we are doing the fifth part. If I put x one minus two, y is also one which is
24:13equal to four, one minus two equal to four minus one equal to four, but it is not equal. So that
24:21means this is also not a solution. Okay, so just put the values in the equation and you'll get to
24:28know if both the sides are equal. It is a solution otherwise not. Moving on to the last question. Now
24:36question number four, you have to find the value of k if it is given x is equal to two y equal to
24:44it is a solution. If these coordinates are the solution, this should satisfy this equation.
24:53So what you will do put x equal to two and y equal to one in this equation and just find the value of
25:01k. So it is two into two plus three into one that is given to you as equal to k two into two is four
25:11plus three into one is three which is k four plus three is seven. So what is the value of k
25:18it is given as seven. Right? I hope you found this session useful. If yes, please don't forget to see
25:26another session where we'll be doing some very very different and interesting topics, something
25:32ahead of what we have already done. Thank you so much for watching it and please don't forget to
25:38like and subscribe. Take care.