+619 Let $x$ and $y$ be real numbers whose absolute values are different and that satisfy begin{align*} x^3 &= 20x + 7y \\ y^3 &= 7x + 20y \end{align*} Find $xy.$
+129852
x^3 = 20x + 7y (1)
y^3 = 7x + 20y (2)
Add (1) and (2)
x^3 + y^3 = 27x + 27y
Factor both sides
(x + y) (x^2 - xy + y^2) = 27 (x + y)
(x^2 - xy + y^2 ) = 27 (3)
Subtract (1) and (2)
x^3 - y^3 = 13x - 13y
Factor both sides, again
(x - y) (x^2 + xy + y^2) = 13 ( x - y)
(x^2 + xy + y^2) = 13 (4)
Subtract (3) from (4)
2xy = -14 divide by 2
xy = -7
