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

Ajouté : 2026-06-02

[00:00:00]In this video, let us solve for x and y.

[00:00:09]Given x + y is = 100 and x * y is also = 100.

[00:00:30]Let us call this equation one and call this equation two.

[00:00:38]Then from equation one if we solve for y we get 100 - x.

[00:00:45]So we have y is = 100 - x. I'm going to call that equation three.

[00:00:57]Now if we go back into equation two and make a substitution for y using 100 - x then we get x into 100 - x then equal to 100.

[00:01:19]Next we open up this bracket. X * 100 will give us 100 X.

[00:01:25]Then - X * X. This will give us X^2 is = 100.

[00:01:34]Or we can rearrange this to give us -x^ 2 + 100x is equal to 100.

[00:01:46]I'm going to move this to the left so that we have -x^2 + 100x - 100 is = 0.

[00:02:01]Now because the coefficient of x squ here is negative I can multiply through by -1.

[00:02:13]So negative * negative here will give give us positive. So we have x^2 this becomes negative 100x and this becomes positive 100 then equal to z.

[00:02:31]And now we have a quadratic equation.

[00:02:34]Let us solve for x.

[00:02:37]To begin, I'll first compare this with a x^2 + b x + c is equal to zero. This will allow me to get the coefficient a and b and the constant stem c which will be 1 and 100 respectively.

[00:03:06]Then to solve for x we use the quadratic formula.

[00:03:10]x is = tob + or minus square<unk> of b ^ 2 - 4 a c over 2 a.

[00:03:26]So all we need to do is just plug in this values into this formula.

[00:03:32]This will then give us x = minus b itself is negative. So we have - 100 plus or minus square<unk> of -00^ 2 - 4 * a is 1 and c is 100.

[00:04:06]Then all of that divided by two because a is one.

[00:04:18]This will then give us x is = positive 100 plus or minus square root of we have 100^ 2 - 4 * 100 then divided by 2 * 1 is 2 giving us x is = 100 plus or minus square root of 100 is common here. So let's do 100 into brackets 100 - 4 then / 2 x is = 100 + or minus square<unk> of 100 * 100 - 4 is 96. six.

[00:05:28]So we get x is = to 100 plus or minus. If we separate this we get square<unk> of 100 *<unk> of 96 then divided by two.

[00:05:49]So that this gives us x is = 100 plus or minus square root of 100 is 10.

[00:05:58]If we look at 96 this is 16 * 6 then divide by two.

[00:06:13]So we get x is = 100 + or - 10 *<unk> 16 *<unk> 6 then / 2 giving us x is = 100 + or minus square root of 16 is 4 4 * 10 is 40 then <unk>6 / 2.

[00:06:49]I'm going to separate this division to give us x is = 100 / 2 plus or - 40 <unk>6 / 2. 2 here is 1. 2 here 50 2 here 1 2 here 20.

[00:07:12]So this will give us x is = 50 plus or minus 20 20<unk> 6 or we can separate this into x1 is = 50 + 20 <unk>6 or x2 2 is equal to 50 - 20 <unk>6.

[00:07:52]And with these two values of x, we can then get the corresponding values of y.

[00:07:57]Recall from equation three, we had y is = 100 - x.

[00:08:08]Therefore, y1 is going to be 100 - x1. And we know x1 is 50 + 20<unk> 6.

[00:08:21]This will imply that y1 is = 100 minus 50 + 20 <unk>6 which will give us y1 is equal to 100 - 50 here is 50 - * + is - 20 <unk>6 six.

[00:08:51]Similarly, we're going to have y2 = 100 - this will be 50 - 20<unk> 6 and then we get y2 is equal to 100 - 50 here will give us 50 - * - is plus. So we have 50 + 20 <unk>6 giving us every possible values for this problem. Now let us do a quick check to confirm that these values are correct.

[00:09:30]To check, we need to substitute each of this into the given problem, which was x + y = 100 and xy also equal to 100.

[00:09:48]We can use either of these solution sets to check for correctness. I'm going to use the first one. Now starting with the first equation, x + y is = 100. This will then imply 50 + 20<unk> 6 + 50 - 20<unk> 6 to give us 100.

[00:10:25]So this here cancels this and then we have 50 + 50 that will give us 100. So we have 100 is equal to 100. And because the left hand side balances the right hand side confirms that this solution is correct.

[00:10:43]But we need to also check with the second equation. Putting this values.

[00:10:48]Therefore, xy is = 100.

[00:10:57]This will then imply 50 plus 06 * 50 - 2006 to give us 100.

[00:11:22]50 here * 50 here will give us 2500.

[00:11:28]Then we do 50 here times this. This will give us - 1,6.

[00:11:38]And this here times this will give us plus 1, <unk>6.

[00:11:45]And finally this here times this here because the signs are different. So we have 20 <unk>6 raised to power 2 to give us 100.

[00:12:08]This is 2500.

[00:12:13]This and this take care of each other cuz they have opposing signs. Then we have - 20<unk> 6 raised to power 2 to give us 100.

[00:12:29]Here this is 2500 minus by law of indices this will give us 20 raised to power 2 * <unk>6 raised to power 2 to give us 100.

[00:12:47]This takes care of this.

[00:12:49]So we have 2 500 minus this is 400 * 6 to give us 100.

[00:13:01]So we have 2 500 minus this will be 2 400 to give us 100 and this is 100.

[00:13:15]So 100 is equal to 100. Again the left hand side balances the right hand side.

[00:13:21]Also confirming that these solutions are perfectly correct.

[00:13:28]Thanks for watching. Please like and share and also remember to subscribe to my channel and I'll see you in my next video. Bye.