Let x and y be real numbers whose absolute values are different and that satisfy

I will assume you want to find  x,y   where x and y have different absolute values...(NOT   x*y as written in your question)

  I used a graphical approach (desmos) to graph the two equations you will see the following sets of coordinates satisfy the equations :

x,y  = (-1.7 , 4.1)     (-4.1, 1.7)   and   (4.1, -1.7)  and (1.7, - 4.1)

0,0    -3.6, 3.6   and  3.6, -3.6     5.2 , 5.2     -5.2, -5.2 also work, but the absolute values of x and y are equal at these points, so they do not satisfy the question.

Here is the graph:

 I believe this problem has appeared several times before....but, anyway

x^3 = 20x + 7y

 y^3 = 7x + 20y     first, add these

x^3 + y^3  =  27x  + 27y

(x + y)( x^2  - xy  + y^2)  = 27(x + y)   divide by x + y

x^2 - xy  +  y^2   = 27   (1)

Next, subtract the first two equations

x^3 - y^3   = 13y - 13y

(x - y) (x^2 + xy+ y^2) =  13(x - y)   divide by x - y

x^2 + xy + y^2  =   13   (2)

Subtract  (2) from (1)

-2xy  =  14        divide both sides by  -2

xy  = -7

No problem ACG....I wouldn't expect anyone to scroll through the thousands of questions on here to see if theirs has been asked before.....I just happened to remember this one....to give full credit, I believe that heureka  or alan may have solved it originally....

Oh...Ok    sorry....    If you multiply all of my x.y 's together (rounded)  they come out to -7