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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 $xy.$

michaelcai  Oct 12, 2017
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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

 

 

cool cool cool

CPhill  Oct 12, 2017
edited by CPhill  Oct 12, 2017

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