Solve for x and y in this nice simultaneous equation | Math Olympiad Mathematics

[00:00:01]In this video, we want to solve for X and Y.

[00:00:08]Given X + Y is 40 and X * Y is equal to 80.

[00:00:15]Let us call this equation one.

[00:00:18]Then this will be equation two.

[00:00:23]Now, from equation one, I'm going to make Y the subject of the formula so that Y will now be equal to 40 - X.

[00:00:40]Now, looking into equation two, in place of Y here, I'm going to put 40 - X.

[00:00:47]Then I have X into 40 - X is equal to 80.

[00:01:01]We need to carefully open up this bracket.

[00:01:03]X * 40 will give us 40 X.

[00:01:08]Then X * X here will give us X squared.

[00:01:14]Then equal to 80.

[00:01:18]We can rearrange this to give us negative X squared + 40 X.

[00:01:27]Then bring this 80 to the left as negative 80.

[00:01:32]Then equal to zero.

[00:01:36]From here, I'm going to divide through by negative one on both sides.

[00:01:47]So that this becomes positive X squared - 40 X + 80 is equal to zero.

[00:02:03]This brings us to a quadratic equation, and to solve this, we first need to compare with the standard quadratic equation ax squared + bx + c = 0.

[00:02:17]This is so that we can get value for a, b, and c.

[00:02:21]a b being one which is the coefficient of x squared here.

[00:02:26]b being -40 which is this, and then c being 80.

[00:02:34]We're now going to apply the quadratic formula to solve for x, which will be -b + or - square root of b squared -4 * a * c then divided by 2 * a.

[00:02:59]And since we have these parameters we're just going to plug it into this formula to get x is equal to -b which is -40 then + or - square root of -40 squared -4 * a a is one, and c is 80.

[00:03:30]Square root of that then divided by 2 * a a is one.

[00:03:43]This will give us x is equal to this will be positive 40 + or - square of this will be 40 squared minus 4 * 1 is 4. 4 * 80 here We can still leave this as 4 * 80.

[00:04:11]Then divided by 2 * 1 is 2.

[00:04:18]Let us do this as x is equal to 40 plus or minus square root of 40 squared is same thing as saying 40 * 40 then minus 4 * 80 here is 2 * 40.

[00:04:43]Then divided by 2.

[00:04:47]This will give us x is equal to 40 plus or minus square root of So, 40 is common here.

[00:04:58]We can do 40 into What I'm left with 40 minus 4 * 2 is 8.

[00:05:06]Then square root of that divided by 2.

[00:05:13]Giving us x is equal to 40 plus or minus square root of This will be 40 * 32.

[00:05:28]Then divided by 2.

[00:05:32]This will give us x is equal to 40 plus or minus I'm going to write this as 40 here is 4 * 10, but I'll write it as 4 * 2 * 5 * 32 then / 2 Now X is going to be 40 + square here is 64.

[00:06:10]So together with this we have 4 * 64 then * square root of 5 then divided by 2 This will give us X is equal to 40 + So square root of 4 is 2.

[00:06:35]Square root of 64 is 8.

[00:06:39]Then root 5 / 2 which would then give us X is equal to 40 + - 16 root 5 / 2 Therefore X is now equal to 40 / 2 + - 16 root 5 / 2 2 here is 1. 2 and 16 is 8.

[00:07:19]2 here is 1. 2 and 20 and 40 is 20.

[00:07:25]So X is equal to 20 + 8 root 5 And if we separate this we get X1 is equal to 20 plus 8 root 5.

[00:07:46]And x2 equal to 20 minus 8 root 5.

[00:07:57]So, this gives us two possible values for x and now we need to get two corresponding values for y.

[00:08:05]Earlier in the video, we got y is equal to 40 minus x.

[00:08:12]This would then imply that to get y1, we're going to do 40 minus x1. And similarly for y2, we're going to do 40 minus x2.

[00:08:25]So, for x for y1, we're going to have y1 is equal to 40 minus 20 plus 8 root 5.

[00:08:41]This will give us y1 is equal to 40 minus 20 is 20.

[00:08:47]Minus times plus is minus.

[00:08:50]Then 8 root 5.

[00:08:54]So, y2 is going to be 20 plus 8 root 5.

[00:09:03]Since the solutions appear to be symmetrical.

[00:09:07]Given us all four possible solutions once to this problem.

[00:09:13]Thanks for watching. Please like and share. And also remember to subscribe to my channel. And I'll see you in my next video.

[00:09:21]Bye.