| X and Y |
| If XY = 10 and X2Y + XY2 + X + Y = 99, find X2 + Y2 |
Problem Moderated by: Douglas |
| Problem Solution |
For many, the instinctive reaction may be to try to find the values of X and Y. However, since the problem never asks for those values, we should wonder: Is it possible to find the sum X2 + Y2 without actually finding X and Y? The answer is yes.
Beginning with X2Y + XY2 + X + Y = 99, we rearrange the terms to arrive at:
XY(X + Y) + (X + Y) = 99
(XY + 1)(X + Y) = 99
Since XY = 10,
11(X + Y) = 99
X + Y = 9
Now take this last equation and square it, giving:
X2 + 2XY + Y2 = 81
X2 + Y2 + 2(10) = 81
X2 + Y2 = 81 - 20 = 61
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