 Hello friends. So in this session, we are going to take up another method of solving a pair of linear equations in two variables Namely the cross multiplication method. So I have already written down the pair of linear equations in standard form a1x plus b1y plus c1 equals 0 and a2x plus b2y plus c2 equals 0. Now, please note that all the terms are Mentioned on the left-hand side of the equality sign Right. So this is the first criteria of whatever technique we are going to discuss if you have another Format that is if you're writing the constant terms on the right-hand side That is let us say if you're talking about equations like 3x plus 5y equals 2 and so forth Then the technique will be slightly different. Okay So in this technique what we are going to discuss, please note All the terms should be on the left-hand side. Otherwise, you will make mistakes So please make sure that all the terms are also I'm going to write it here. So please make sure that all the terms are written in left-hand side Once this criteria is fulfilled, then we can proceed. So what is the method? I'm now going to describe it to you. So let us say these are the two equations So what you need to do is you write x, y and 1 like this Then you write the columns of numbers which I'm going to write like this So we write b1, b2 and then you write c1, c2 Then you write a1, a2 and again you write b1 b2. Did you notice? So what is that we have started with? When I have written x over here, I've started with the column next to x So b1, b2. Then whatever is next c1, c2 and then I have gone back to a1, a2 And then finally to b1, b2. So four such columns will be made. Okay Then please remember that you have to multiply like this top left right bottom first Okay, so I will write it here top left into bottom right right and then you have to do what? second thing is bottom left bottom left into top right This is the sequence. Please remember. So top left bottom right first right and then then same thing will So in this case also, you will go like this and then like that and in this case like this and Then then like that. What are these arrow? Basically mean so hence now the second step is x and Then right under the bar. So what did we multiply first top left bottom right? So write that first so b1, c2 and then put a minus sign and then write b2 c1 Okay, then equate it to the next term that is y And again go for c1 a2. So this is c1 a2 top left bottom, right? minus c2 a1 Okay, and then third term will be simply one Right a1 b2 minus a2 b1 if you notice carefully Underneath x there is no a that is a coefficient of x underneath y There is no b that is coefficient of y and underneath underneath one. There is no talk of c That is a constant term, right? So this is how this is also you can you know, so Remember like this. So what how to remember? So let us say x So x coefficient is a so a will not be there. So simply write bc minus bc Simply write this and then you have to start with one go to two then two and one reverse Right, then underneath y. What do you need to write? Everything but the coefficient of b. So what is coefficient of b? C and a mind you c will come first not a because always remember This is also handy to remember. Let us say a b and c this goes in clockwise cycle So a will be first then b if a b are together That is the order will be a b and if b and c are together the order will be bc If c and a are together the order will be c a right so when c a are together c will be written first So c a minus c a you write and again you write one two two one and then third is One and then again constant term So hence constant term will be missing. So hence you have to write a b minus a b in that order and then one two two one order is important guys, so please Take note of it, right? So the moment you have got this Then your job becomes very easy. Why so do you get equate these two first and third? Then you'll get x. Let us do that. So what is it x upon b1 c2 minus b2 c1 is equal to 1 upon a 1 b2 minus a2 b1 so this implies x will be equal to b1 c2 minus b2 c1 upon a1 b2 minus a2 b1 why because this thing Goes on top here cross multiplication and divided by the denominator Fair enough. So let us also find out why this in the same way So why upon what is it c1 a2 minus c2 a1 is equal to 1 upon a1 b2 minus a2 b1 Right, how do we arrive at it? So basically we equated these two Okay, so what will I get I will get I Will get why again same method c1 a2 minus c2 a1 divided by a1 b2 minus a2 b1 This is cross multiplication method If you see directly if you remember the formula and if you have all the values of a1 b1 c1 and a2 b2 c2 You can deploy those values in these two formula and you'll get the value of x and y. Let us take an example Okay, so let us say our equations are 2x plus 3y minus 5 equals 0 so this is equation number 1 and Equation number 2 is 3x minus 2y plus 2 equals 0. So let us write down all the values. So a1 is 2 clearly b1 is 3 and c1 is minus 5 Many people leave the sign here. So please be careful c1 is minus 5 and not simply 5 similarly a2 is 3 b2 is negative 2 and c2 is 2 Right. So now that we have all these values. Let's deploy them. So what is x directly? You can find out b1 c2 That means 3 into 2 minus b2 c1. So b2 is minus 2 and c1 is minus 5 and then that divided by a1 b2 is 2 into minus 2 and then minus a2 b1 So it is 3 into 3 Okay, so hence it will be 6 minus 10 by minus 13 which is nothing but 4 upon 13 now what is y? y is c1 a2 here y is this much c1 a2 so negative 5 into 3 minus c2 a1 which is 2 into 2 divided by again 2 into minus 2 that is a1 b2 minus a2 b1 3 into 3 if you notice the denominator of both x and y are same Correct. So hence what is the calculation final calculation? It is nothing but minus 19 upon minus 13 So hence 19 upon 13 so this is y and this is x so directly by application of the formula We got the values of x and y this method is called cross multiplication method