 You can see the first question on the board. So try and solve, try solving the first one. So it is the solution of a linear equation is not affected when so basically they are talking about solution of the linear equation all of you have. So just let me give you the basics of linear equation once again so that there is a good recap. So linear equation are of the form of a x plus b y plus c equals zero mind you. We are dealing with linear equations, linear equations in two variables, linear equations in two variables in two variables. So the general form, this particular form is general form. This is general form. Okay, now the question is, there's asking solution of a linear equation, what is solution of a linear equation so solution of a linear equation. Okay. So you have to also take care of one more thing that a should not be equal to zero and B should not be equal to zero for the linear equation in two variables. Okay, so this is the bare minimum requirement. Okay, now solution of a linear equation means those values of. Now, so solution of a linear equation those values of variables x and y which satisfies which satisfy the given equation. Okay, so example let's say I have x minus y is equal to two. Okay, so yeah, so can you tell me guys what are values of A and B and C in this. You can type it in the chat box, let me see whether you are familiar with this what is the value of A what is the value of B what is the value of C in this case. If you compare this with this one. Yeah, answers please you can just type in in the chat box. So someone says 110. Brownhouse says one minus one C is equal to minus two. Yes, so hence 110 is not correct why because this can this equation can be written as x. One times x plus minus one times y minus two is okay so hence what is the value of a is equal to one here so if you compare these two equation this equation with which equation. A x plus B y plus C equals to zero so if you compare a becomes one B becomes minus one and C becomes minus two. I hope this is clear to everyone right this is what is meant by linear equation in one variable and what are the values or what is the what is the value of these coefficients. So the question is, before we do it so NPS score among the folks have you done linear equations. One of yes, yes or no. No, sir. Okay, Indra Nagar guys anyways you haven't done I believe. So never mind. Let me just Roger G. Nagar guys you have done today itself correct. Yeah, so let me just give you a brief intro to linear equation and then I was always having impression that you are you have already done. Never mind. So guys in your till eighth grade you had done linear equation in linear equation one variable. This particular expression appears to be familiar. Now there are few parts to it why it is what is meant by one. These are the things you must know. Okay, now. So what is linear equation so linear equation earlier till eighth grade. You would have done this. You would have done X plus two is equal to five. So what is the solution to this. So you know you have you know how to solve this this equation means five minus two is equal to three. Right. This is what you have already done in the previous grades. Now, give me a moment. Okay, now, in the ninth grade. We have celibates we have linear equation. Okay, so linear equation in two variables. This is what is the subject matter. Okay, so first of all. Yeah, harsher than you're saying something please unmute and say harsher than you want to say anything please unmute and you can say either equation into variables. So, so what is an equal first of all you have studied about algebraic. Let me start with very basic what is an equation. What's an equation so equation is there is a balance and you would have seen this in your previous years, wherein you weigh in something here and counterbalance it with something here. So, you say when the this beam is horizontal we say these are these two weights are equal, isn't it. So here in mathematics equation is there is a quantity one, it's liquid to quantity two. So when one quantity is equal to quantity two, we call it an equation. Now, equations can be of multiple types. So, you have studied algebra. So we can have equations related in algebra and they are called algebraic equations. What does it mean algebraic equation. So here it will be algebraic algebraic expression on one hand expression one is equal to algebraic equation to expression to sorry. It's very, very clear what is meant by expression and what is meant by equation. So here I'm talking about algebraic expression one, which I can write as let's say p of x and as a break expression to which I can write as q of x. Now, you are familiar with these terminologies. What is p of x that means this is an expression in x. That means the variable is x and this one is another expression whose variable is x again. So let's say 2x minus y is not an expression next to x minus five. Let's say 2x minus five. This can be written as px. And qx let it be 7x square plus 4x plus two. Okay, so hence when you equate these two, it becomes an algebraic equation. Okay, so hence algebraic in this case will be 2x minus five is equal to 7x square plus 4x plus two. So what is an equation equation is nothing but quantity one equated to quantity two becomes an equation. And in this case we are dealing with algebraic equations only. So hence you'll have algebraic expression in LHS for hence that this is the form of an equation LHS and RHS we say. So everything on the left hand side of the quality sign is LHS and everything on the right hand side is RHS. Now these expressions could be algebraic example 3x plus five or root 2x square minus 9x. These are algebraic expressions. These expressions can also be trigonometric. So you'll be studying trigonometry in a while in some time now. So hence trigonometric. So for example, sine theta or core second, cos of x or something like that where x is the angle. So trigonometry. So these are the expression trigonometric expressions or let's say sine of x square plus five something like that. So don't get panicked by looking at these you'll be coming across all this very soon. Then there is something called logarithmic expressions. Logarithms also you are going to study in this great sense. It is log of let's say x square plus five. Right. So this is logarithmic. Then there could be exponential, exponential expressions like two to the power x three to the power y and so on and so forth. If you see here, the base is constant and the variable. The index is variable or the power is variable. Hence this is different from x square where variable is the base and constant is the power. Okay, so these if you equate any of these equation expressions with let's say you write expression one on one side expression one on one side and expression two on the other side. So expression one and expression two. So when you do this, you get an equation. And one type of that is algebraic, algebraic equation. So when is algebraic equation when is this called algebraic expression equation when you have algebraic expression is equal to another algebraic expression. Right. And one special case is linear expression on one side and linear expression linear expression. On the other side. So one and this is what we are going to deal with. Now comes the definition of linear expression. So let's talk about linear expressions first. If any doubt, please either you can chat in the chat box or you can tell me by unmuting yourself. Okay. Yeah. So please. Okay. Now, now so linear expression, what is linear expression guys. So linear expression will be of the form of let's say I am first of all, I'm taking linear expression in one variable, linear expression in one variable. So hence the expression must consist of only one variable. Let's say the variable is X. So hence, let's say three X minus five. If you see the power of X here is one. Okay. So hence, these type of expression will be called linear expression. So hence, generalizing generalizing this, it will be of the form of a X plus B. Okay, this is nothing but general form. General form of linear expression, why is it called linear will come to it in a while linear expression expression again I'm reemphasizing its expression we are not talking about equation right now. So a X plus B is of the general form of linear expression where you cannot have a zero because if a zero, then this variable or this part just disappears so hence a must not be zero and B is a constant, any constant. And we are talking about a and B being real numbers. Right, so example of linear expression in one variable are so examples of linear equation in one variable are 2x minus five 3x root 2x. And in, and in these cases, the variable is X, then three by two X plus five. Okay. Now, you will tell me which among the following is linear expression. So one root to Y minus four to three X Y for three root X out of one, two and three, which are which is which one is linear and which one is not linear answer you can key in in the chat box, which one is linear, which one are linear and which one is not linear. So people are saying all are linear. I need to say only one is linear case of says one and two are linear runoff says one and two are linear one and two. Okay. Three is not linear. Four is four. Sorry, I'm writing I have written for my mistake. This is three. All or none. No. So the answer is only this one is linear. This is not linear. This is not linear. Why now focus on this. Why is three X Y is not linear. Whenever you are evaluating the power or the degree. So we are now talking about degree. So I think in polynomial chapter you'd have studied degree. So degree is nothing but highest power, highest power of all the terms, all the terms. Okay, so in case of three X Y. The power is one plus one that is to so here the degree is to because the term is if there is there are you know, in the term there are two variables then the power of that term will be nothing but the sum of the powers of the variables. Another example. So let's say two X square Y. Its degree is three. Why, because it is two plus one. Another example X to the power five Y square Z to the power three. The degree is 10. I hope this is clear. Okay. Yes, you're correct. So this is this is what is meant. So hence if you see here, the power was So hence it is not linear. Here the power of X is half. Right, not equal to one again. This is also first of all this is this doesn't fall under the category of polynomials itself. So but anyways we'll come to that later but here degree is not equal to one. So it's not linear. Okay, so we'll give you some more example. Let me see whether you are able to Now I'm writing giving you more examples. Tell me which among the following are linear one two p square minus five p two three X plus four three seven by two X plus one by X for root two X X square minus four X four divided by X which one is linear amongst all of these only two. What about what about four four is not linear. Yeah, so if you see this is clearly not linear why the power is Two, so hence not linear. This is definitely linear why because power is one. Here if you see X is having a negative power that means it's this this expression can be written as seven by two X plus X to the power of minus one. Now this is creating the trouble. Right, so hence it is not despite the fact that it, it also has another term with power one. This is not. Now this one right now is not in the form of linear expression but it can be reduced to one if X is not equal to zero. If X is not equal to zero this particular thing can be written as root two X minus four and then it becomes a linear expression. So I hope it is clear to everyone what a linear expression is just omit X cancels only when X is not equal to zero otherwise cancelling X is not allowed. Okay, once you're clear with linear expressions then I said linear expression one that's a linear expression number one is equal to linear expression number two the moment it happens. The moment you do this such that such that coefficient of I hope you know what coefficient is so coefficient of coefficient of the variable. Variable in expression number one should not be equal to coefficient of the variable in expression number two. If this is this condition is satisfied then you get a linear equation from two linear expressions example five X minus three five X five X minus three is one expression let's say linear expression. Another linear expression is QX let's say three X minus four. Okay, so when you equate them to equate them you'll get linear expression. This is now an example of linear equation. Now I have changed expression to equation minded in one way in one variable. Now what does this condition mean check what is the coefficient of the variable in expression number one. This is five what is a coefficient of X you know variable in expression number two this is three so five definitely is not equal to three and it satisfies this condition hence this is a linear equation in one variable. Now what if the expression has same coefficient so let's say if you would have written five X minus three is equal to five X plus four this is expression number one expression number two or expression number one to whichever way so in this case you would have proved minus three is equal to four. This is a blender. Okay so hence mind you the coefficient must not be equal. Now, once you have this kind of an equation you can always simplify and write it as five X minus three minus three X plus four I have just taken these two terms on the left hand side and then equate it to zero and then you get two X plus one equals zero if you check guys this is of the five X minus three X plus four. This is what is called linear equation in one variable. Okay, so this is what is linear equation in one variable now let's move ahead and come to your level and this is nothing but you will be studying linear equation in variables. So as this only X was not sufficient we are increasing our headache by increasing one more variable. Okay, linear equation into variable say here the game starts again to use you know it has to be expression one is equal to expression two. So this is a equation now now we have to just make sure that here expression is expression with two variables and that to linear expression with two variables example. P x y now I'm introducing another variable this is just a notation P x y which is equal to let's say two X plus three y minus four this is an example. Okay, so a linear equation linear expression expression into variables will be of the form of a X plus B y plus C. Right now X and Y could be D and U or S and V whatever so let's say you it can also be a U plus B V plus C where you is you and we are the variables these are the variables. Okay, so it need not be only X and Y okay when you have two such expression guys. And you create them you get a linear equation into variables let's say let's start by taking example P x y is let's say two X plus three y minus four the previous one which we took and Q x y will be equal to three X minus four y plus two. Okay, these are the two. Right now someone is asking what is the difference between expression and equation. Okay, now as I told you expression is just so if you have to just let me deviate a bit and clarify this doubt expression, what is expression expression is nothing but collection of. Collection of collection of algebraic in this case we are only restricting ourselves to algebraic algebraic terms now I hope you're clear with what is algebraic terms separated by separated by plus or minus name. Example. Two X plus five is an express is an expression three X minus four Y is another expression for X square plus two X Y plus three Z is another expression, all are different different expressions. Okay, so when you so these are expressions but how is it different from equation now the moment you take to expression and put a equal design in between so let's say two X plus five is equal to three X minus four Y this becomes in this is an. I hope this is clear to you. Okay, is that clear to the person who asked the question if you have you can again post. Right, so coming back to this so when I am equating these two so two X plus three Y minus four is equal to three X minus four Y plus two. Okay, so hence this is our equation linear equation into variables, yes obviously not in a standard form but you have just achieved an equation. Now, the best idea is to X you know simplify this so to X plus three Y minus four is that left hand side and whatever is there on the right hand side please take on to the left hand side and it becomes minus three X plus four Y minus two and whatever is left on the right hand side is. Zero, nothing is on the right hand side. Okay, so hence now simplifying it what do you get to X minus three X you know by your concepts of algebra earlier it will be minus X plus seven Y minus six zero. So this is now you are now dealing with linear. Equation into variables right so generalizing it guys generalizing. So what is linear equation into variables so linear equation in two variables will be of the form will be. Of the form what form. AX plus BY plus C equals zero. Now you'll ask me what is a B and C you clearly X and Y are variables isn't it so this X and this Y are variables. Okay, and what is a B and C so a B and C are. Real numbers any real numbers you can think of. Okay, what is it no crosstalk here yes. Okay, so only when there is a question you will post it in the chat box otherwise do not do that. Okay, now ABC are real numbers and a should not be equal to zero be should not be equal to zero why is this condition valid because so. So ABC are real numbers a not equal to zero be not equal to zero. Okay, so so hence this is a condition and I'm saying why is the why are these conditions important because if a and B become zero then you get. If let's say a and B both are zero what will you get you'll get zero plus zero plus C is equal to zero now a non zero constant cannot be equal to zero. So what does it mean what does it mean a non zero quantity cannot be equal to zero so hence and you are basically defined if you basically. Define the equation itself is not defined then understood now so you now know what are linear equation into variables so questions of in your textbook would be of this form. Okay, question on the textbook would be of this form so let's say 3x minus root 2 y plus four equals to zero find a B and C quick yes tell me what are the values of a and B and C quick three minus root two and four is great great so all of you are getting it now question number. Find the again in x by two is equal to y by three plus four question number two, what are the values of a B and C, what are the values of a B and C, what are the values of a B and C, so a is half B is minus one by three and sees for very good so what do you need to do is, you can just express it like this, you take everything on the left inside it becomes y by three. And becomes minus four equals to zero, so now it matches with our expression or form general form a x plus b y plus C equals zero. So clearly a is one by two B is minus one by three and C is clear very good question number three find find a B and C yes. Very good. Correct. So what is this so you can again write it as x minus root 2 y minus root 2 y plus zero is equal to zero so if you again match this with this x, you know form then a is equal to one B is equal to minus root two and C will be equal to zero. Very good. Next is, next is what question number four x is equal to minus nine find a B and C, find a B and C yes so it is lots of you are correct so it is nothing but x plus one times x plus zero times y plus nine equals zero. So hence a equals one B equals zero C equals. Okay great. Now do this question. Now question is the cost of cost of ball pen ball pen is rupees five more than more than half of. Half of half of the cost of a fountain pen. Okay, so can you convert this into an equation. The cost of ball pen is rupees five more than half of the cost of a fountain pen. Express this into linear equation, linear equation in two variables. Yes, so how to approach such yes most of you are approaching in the right direction but see this is how it is done. First of all find where is terms like is R was and all that here there is is so this is is is is your equal to sign. So here the is is equal to sign. Okay now the cost of ball pen so it is always advisable to pick variables so that you can correlate to so ball pen I will write. P or BP or be whatever be cost of ball pen is be okay and cost of ball pen is be then next keyword is is so you write is five more than so five more than means plus. So you write five plus half of the cost of a fountain pen so half of so half and off means multiplication so half in two. Okay, so this is so hence this is so if you have to you know most of you have taken. B as X so hence it is X is equal to five plus half why so rearranging so X minus half why minus five is equal to zero. Okay, and hence you can simplify further by multiplying the entire equation by two. Okay, so you must learn all this multiplying the entire equation by two you will get 2x minus y minus 10 equals zero. Okay, and this is what the first question is related to here if you see says it says what does it say the solution of a linear equation is not affected when. The same numbers is added to both sides of the equation. We multiply or divide both the sides of the equation by the same non zero numbers. We add a number. We add a number. To one side and subtract the same number from the other side of the equation both a and b so answer is D understood. Understood. So hence what did we learn we learned that if there is an equation let's say a x plus v y plus c equals to zero. Okay, now if you add any. Okay, so now so if you see a x plus v y plus c plus let's say two will be equal to two so I'm adding two on the both sides. Now next is let's say I can subtract as well so let's say v y plus c minus five is equal to minus five both sides. Yeah, this is fine. And similarly third one is you can multiply and divide by non zero number non zero you cannot divide or multiply by non zero non zero. Okay, so non zero number has to be multiple you can use non zero number for multiplication and division or multiplication and division multiplication or division so you can do that for example if three x minus five y plus four equals zero you can multiply or divide by let's say the entire equation gets multiplied by seven let's say. So what is it it will become 21 x minus 35 y plus 28 equals zero. You can also divide and do multiple other. Right now the other concept which you must learn is solution solution of a linear equation and the second question is based on this if you see solution of a linear equation. Yes, the question is which of the following is not a solution of what does it mean what is meant by solution of a linear equation so hence let's take an example x plus y equals. Okay, this is one. I need to find out values of x and y such values of x and y now these so far we were discussing or we were taking x and y as variables now I am interested in finding out the value of x and y such that. The equation is satisfied so let's say just let's try some trial and error method so x equals to zero let's say and why is equal to one if this is the case if I put it back in LHS you see LHS becomes zero plus one is equal to one which is. Yep, so this is what it is. So hence we say x equals to zero and why is equal to one is a solution solution. So those values of x and y which. Satisfy. Given. Question those values of x and y which satisfy the given equation so guys can you tell me some more values of x and y which will satisfy this x plus y is one any other any other solution to it so once one solution is zero comma one where x is zero y is one. So someone says minus one. Yeah, very good 1000 minus 999 and there could be any anything and you know lots of lots of solutions so for example one comma zero is another solution to minus one. Minus one. Minus one. To is another solution and so on and so forth right so if you see there are infinitely many solutions infinitely many solutions so you can have one million. That is equal to 10 to the power six let's say x equals to 10 to the power six and why is equal to 10 to the power six minus one. Right no sorry and why is it yeah 10 to the power sorry it will be this like that if you do this then you will get. X plus y is equal to one in all these cases correct so basically. Lots of such. Solutions so what do we what do we learn the learning is there are. Learning what is the learning from this exercise. There are. In fine nine. The infinitely many infinitely many many many please emphasize many solutions infinitely many solutions to a linear. Equation in two variables. Okay and how to find them so how to find them let's let's try and see so let's say let's take a random equation like 2x minus 3 y plus 4. Equal zero and we are interested in finding the solution to this equation so what you need to do is you express. One variable. One variable. One variable. In terms of. In terms of the other okay how to do that so let's say you keep one variable on one side and everything else on the right hand side okay so let's say I am keeping isolating 2x on the right left hand side and everything goes on the right hand side becomes 3 y minus 4. You can do the do it other way around as well you can keep minus 3 y here and send 2x on the right hand side so it becomes minus 2x minus 4 both are. And you will get the same solution actually friends if you see what is x now so as we learned we can divide or multiply the entire equation by the same number so hence I can multiply it by half and why am I multiplying it by half because then. x will be divided off its. Its position so if I do that I'll get x is equal to 3 by 2 y minus 2 so this is expressing one variable in terms of the other okay next is. I can do the same thing here I can multiply the entire missing by let's say minus 1 by 3. So what will you get you will get y is equal to 2 by 3x plus 4 by 3. So why plus y equals 2 by 3x plus 4 by 3 now what you can do is you can now take random values of y and get corresponding values of x so let's say when y is equal to 0 in this. This is part of the equation. y is equal to 0 means x equals minus 2 so clearly minus 2 comma 0 is a solution how do I know that I can check how deploy minus 2 comma 0 back to the original equation and see whether it is satisfied so let's check. So LHS is 2 into minus 2 because x was minus 2 minus 3 times 0 plus 4 is actually 0 which is equal to RHS so hence we say minus 2 comma 0 is a solution. Okay similarly you can have many many more how if you put y is equal to 1 here in this in this equation you will get x equals 3 by 2 minus 2 which is nothing but minus half so we get another solution minus half. Okay minus half comma 1 here to one so hence keep keep in mind all these solutions guys we will be doing something now so that you can understand why we call it linear equation so let's try and find out few more few more so. For this expression equation itself let's say we are plotting we are having x and y values corresponding values okay let us put it in a table what x and y values we are getting and we'll see what are we going to use them for. So hence when y was zero here x was minus 2 okay when x was okay so you will see what are we going to deal what are we going to do with this so hence next set of value is x is minus half and this is one okay let's let's see. Another another example so let's say we take y equals minus one in the same this expression y equals minus one in this so you'll get what x equals 3 by 2 into minus one. Into minus one and minus two so hence what will you get you'll get minus three by two minus two which is minus seven by two right so hence it is minus seven by two and minus one please check. Yep, so as it says y equals 3 by 3 and x equals to zero x equals to zero and y is equal to 3 by 4 y is equal to 4 by 3 yes I think you're correct so let me put your solution as well so as it says. x equals to zero and four by three this is given by. Now I am going to do something. Yes, so here is the graph in tune and I will now plot the I have the. I'll be plotting all these numbers so first number was. Okay, so I am plotting it and I get this first point can you see the screen I hope you are able to see the screen guys so minus two comma zero next point was minus half so next point was next point was. Next point was next point was minus half minus half minus half comma one, but I got the second point third point was minus seven by two seven by two comma minus. The third point is also there and fourth point zero zero comma four by three so do you see something guys if you see if I have to join if you see if I just join these with from this point. So hence you see all these points are lying on a line and hence boss they are these kind of equations are called linear equation you can see that if I take any random value on this let's say this point. So four comma four, so someone was saying four comma four so he point if you see here I'm hiding on the left hand side of the screen four comma four okay so let me say this point so you can see seven point nine five six point six three or let's say. Yep, so all these are so if you if you see all these points which are quite plotting all are these are x and y values so all these x and y values will satisfy here is the equation if you see this is a question which we are plotting right so there are you can see for yourself infinitely many infinitely many solutions many x and y many x and y is will be there. Yep, so this is why and all of them are falling in a line hence these kind of equations are called linear equations okay now since it is two variable since it is two variable you are getting a line in a 2D plane. What will happen if you have another variable added to it so what I'm saying is let's say you had this equation 2x plus 3y plus 4z my plus five is equal to zero if you see there is an additional variable added to it third variable. Now this is nothing but a line in 3d space so if you if you have only two variable you have a line in a 2d space if you have three variable you'll be having line in 3d space and if you have more than three variables then we will not be able to visualize this and but then equations in more than four or three or four variables do exist in practical life. Okay so I hope you are now aware of what linear equation is what is the linear equation in two variable and what are the different coefficients a b and c and why are they called linear equation how to find out solution to linear equation. Let's get back to the worksheet and see if you can solve a few more okay so here is the third question. So now we already saw that so let's say what is the third question so answer is clearly the geometrical representation of a linear equation is guys what's the answer. So where are you where is the chat box so yeah so the answer is yeah a very good so all of you are now clear with this okay next question. The points two zero minus three zero and four zero lie. Option is. So two zero two comma zero so let's have this. So let's say we have this so two comma zero where is two comma zero so what is the answer be on the x-axis all are. All are on the x-axis why because two comma zero will be let's say this three minus three comma zero is somewhere here. And four comma zero is somewhere here four comma zero right so all are in x-axis very good next examine if minus two is a solution of two x plus four equals to five answer is yes or no is x equals is minus to a solution. Very good it's definitely not how do I know why because. So does it come zero. Plus four is equal to zero and it's not five that's why. Great solve two x plus two equals 10. So the six. Four four four four four four four four four. For times go to question number seven find the value of x for that is why the equation this for x equals three three three three three three three three three three three three. X equals three very good go to question number eight go to question number eight. So I'm not solving seventh why because let me just solve it for others as well. well 2.4x minus 0.4x equals 6, the difference is nothing but 3x equals 14, answer is 14 or 7? 7 plus 7, so it is nothing but c5 by 2x plus 7 by 2x plus 2x equals 56, so I have just combined all the x together for an easier calculation, so hence 5 by 2 plus 7 by 2 is 6, so 6x plus 2x is 56, so 8x is 56, so x is 7, Sanjana y14, 7, calculation error, do not do calculation error, okay, question number 9. Solve x. Solve 2. 2, how to solve that question? It is a quadratic equation, this is the middle term. Yes, so it can be the middle term. Of both the numerator as well as? Denominator. So numerator, the products are x plus 4 and x plus 1 and denominator products are x plus 2 and x plus 1. x plus 1, x plus 1 cancel, you cross multiply, you get 2x plus 8 is equal to 3x plus 6 and you get x is equal to 2. So x is x plus 1 plus 4, x plus 1 divided by x, x plus 1 plus 2, x plus 1 equals 3 by 2. So x plus 1 brackets, x plus 4 divided by x plus 1 brackets, x plus 2 is equal to, I hope everyone is following it. Now, you are cancelling, there is where there is a catch. When you are cancelling, please write 4x not equal to minus 1. Because guys, if x equals to minus 1, then you cannot cancel 0 by your 0. So please be very, very careful, when you are dealing with this, you have to write this step ideally. x should not be equal to minus 1, if x equals to 1 you cannot cancel, minus 1 you cannot cancel. So let's say for x equals to minus 1, not equal to minus 1. So hence you can now cancel this and you will get x plus 4 by x plus 2 is equal to 3 by 2. So then cross multiply, so hence 2 times x plus 4, so this goes here and this entire thing goes here. So 2 times x plus 4 is 3 times x plus 2, so 2x plus 8 equals 3x plus 6. So dear friends, x equals 2. Yes, go to question number 10. Question number 10 is write 4 solution to this, pi x plus y, question number 10. Pi x plus y equals 9, give 4 solution to this. So 0 and 9. x equals to 0 and 9, be very good, this is first one. Sir and then 1 by pi and 8. 1 by pi and 8, very good. So minus 1 by pi and 10. Very good, minus 1 by pi and 10. One more, so someone is saying 9 by pi. Sir 2 by pi and 7. 2 by pi and? 7, very good. So you are going in that, you now know you can just have any infinite, you can plot this and find. So 3 by pi and 6. Great, so good. Next, find out which of the following equation have x equals to 2, y equals to 1 solution. Sir 1 has. 1 has, how, what do we do? Sir x is equal to 2 and y is equal to 1 and 3. Sir 1 has, 1 and 3. So 3 has, 3 also. So only 1 and 3. So only 1 and 3. 1 and 3, yeah. 1 and 3. How do you do it? You put x equals to 2, y equals to 1 and check whether LHS is equal to? RHS. Question number 12. Proceed, question number 12. Express y in terms of x and x in terms of y. Cool, quick. Question number 12. So x is equal to 8y minus 5 whole by 7. Sir x is equal to? 8y minus 5 whole by 7. 8 by 7, y. 8 by 7, y minus 5 by 7. Minus 5 by 7. And y is equal to 7x plus 5 whole by 8. Y is equal to? 7x plus 5 by 8. 7 by 8x, you have to separate it out and then? 5 by 8. Correct. You got 3 marks here. Next, draw the curve. Very, very interesting. How will you draw this? 3x plus 2y is equal to 6. What is meant by drawing this curve? So again, you have to draw a curve. What do you need? So y is equal to 3. Basically, you need x and y, right? Points. How do you draw a curve? You get points and then what do you do? You just join them, isn't it? So hence, we need to find out points x and y on the graph. So what is given to you? Because it's on the y-axis. So this is a method. So method is, first of all, express one in terms of the other. So hence, can I not say 3x is equal to minus 2y plus 6. And hence, x equals to minus 2 by 3y plus 2. Correct. So let's now draw the table. Draw the table. x and y. Okay. So when y is 0, what is the value of x? 2. Correct. And when x is 0, what is the value of y? 2. No, when x is 0, what is the value of y? 3. Right? So for any linear equation, guys, you just need two points, but you can actually take some more points to draw and let's draw the curve. 2 comma 0, 0 comma 3. So hence, so let's plot the curve, 2 comma 0 and 0 comma 2. Oh, sorry, 0 comma 3. It should be 0 comma. This is 0 comma 3. Yup. And now let's join it. Okay. For Raja Ji Nagar, guys, do you see it is negatively sloped? Yes, sir. And can you see the intercept as well? Intercept. Y intercept is 2. So hence, now you see how it appears when the line is drawn. Yes, sir. Okay, so the slope is negative. Why? Because as x is increasing, y is going down and and this length is this length is very different. So hence, you will see, you can, you know, all of you, I advise, please download this. Download it. And you keep on experimenting with it. For example, let's say if I say 3x plus 2y plus 5 equals 0. Can you see this new line? Yes, sir. What is the similarity and dissimilarity? Sir, one is both are negatively sloped. Both are parallel. Can you see that? Yes, sir. So they won't have any common solutions. They have any common points. Now, if you can check, you can check. What is the observation? Observation here is the coefficient of y of both. Both are different. Both are different. Coefficient of x and coefficient of y are same. Only the constant term is different. Can you see that, guys? Coefficient of x are the same, but constants are different. Yes. So what do we infer? We infer that if any equation, if you just keep the coefficient same and keep changing the constant term, you'll get what? A parallel line. Yes, parallel. You'll come to know when we'll be dealing in the class. So this is observation. So please download and try to change. So let's say if I change now the coefficient of x only, what do I get? So I reduce it to 2x now. So 2x plus plus 2y plus 8 equals 0. Can you see that? What did you see? The lines intersect so they have a constant. Can you see that? Yes. Can you see the slope? It started turning anticlockwise. It started turning anticlockwise. So as the slope is now increasing towards Acha, some few more, the thing is, for example, if x is equal to 6, what is this line? So it passes to the origin because... No, it goes to 6. It will not pass through the origin. It's parallel to the y-axis. Y is equal to 8. Parallel to x-axis. Parallel to the x-axis. Correct. Do you see this? Now, in the morning what I discussed in the class, let's say x is equal to y. Can you see this? x equals y. Where is it passing through? Origin. So if there is no constant term, where does the line pass through? Origin. So let's say x minus 2y. It will pass through. The origin. The next is, let's say, 2x minus 4y equals 0. Can you see that? If there is no constant term, it will pass through. Origin. 6x minus 4y equals 0. See. Download this app, invest some time on this. You can download this on your mobile phone as well. Yes, sir. Just try to change the value of x and y, and I can show you one. Sir, what's the app's name? Y. Yo, Jibra. Yeah, tell me, who's asking what? Sir, when we represent ax plus by plus c in the form of y is equal to mx plus capital C and capital C is not equal to 0, then the line doesn't pass through the origin. Yes. Now, what I am doing, I am changing ac. Slope keeps increasing. Let's first make c equals to 0. When c is going towards 0, I will merge over the origin. Passes through the origin. Passes through the origin. Origin, what is this curve? C is equal to 0. Now, b is equal to c. I am changing d. So, as d increases, slope keeps decreasing. C is the coefficient of y. Slope will decrease. C is the coefficient of y. Slope will increase. See that? Now, c. So, hence, if you increase the coefficient of a, the slope will decrease. Decrease. Let me see that. Inverse correlation. Now, if you keep changing c, c, what happens? It goes up and down. It moves away from the origin. Hence, for different values of c, it will be parallel lines drawn, isn't it? Yes. This is what is the learning. So, one line, simple equation ax plus vy plus c equals to 0. So, please try with your own, you know, app back at home. So, I hope the session was useful to you. We saw one worksheet. In the next, we will solve a few more. And how you are going to approach is, it's always advisable just after the class. If you finish off the exercise, it will help you immediately. Otherwise, by tomorrow, definitely, finish off the equations, all equations exercises. And if you have any trouble, you can always post the queries to me. Is that fine? Yes, sir. The recording of this thing will be available on YouTube in couple of days time. If you have missed the class, you can always go back. Or if you want to go again, the class of Centre Mar will be having more Centre Mar's every day. Bye, sir. Take care. Good night. Bye, sir. Thank you, sir. Bye.