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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 $$xy$$.

Thank you! :)

Dec 22, 2019

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Let xy=k, so that y = k/x.  Substitute this into the first equation to get x^2 - 20x^2 - 7k = 0.

Solve this (using the usual formula) as a quadratic in x squared to get x^2 = 10 plus or minus sqrt(100 + 7k).

The original equations are symmetric in x and y so repeating the procedure will produce an identical result for y.

Since x and y are different, one will have the positive sign in the middle, the other the negative sign.Multiplying the two results produces (difference between two squares): x^2 y^2 = 100 - (100 + 7k) = -7xy.

Since xy is not equal to 0,  it follows that xy = -7.

Dec 22, 2019