SOLUTION: Let x and y be real numbers whose absolute values are different and that satisfy x^3 = 20x + 7y y^3 = 7x + 20y Find xy.

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Question 1051119: Let x and y be real numbers whose absolute values
are different and that satisfy
x^3 = 20x + 7y
y^3 = 7x + 20y
Find xy.
Answer by Edwin McCravy(20060) About Me  (Show Source):
You can put this solution on YOUR website!



Add the two equations:

x%5E3%2By%5E3%22%22=%22%2227x%2B27y

Factor both sides:

%28x%2By%29%28x%5E2-xy%2By%5E2%29%22%22=%22%2227%28x%2By%29

x and y do not have the same absolute values,
therefore (x+y) does not equal 0 and we may divide
both sides by it.

x%5E2-xy%2By%5E2=27

------------------------------------------

system%28x%5E3+=+20x+%2B+7y+%2C+y%5E3+=+7x+%2B+20y%29

Subtract the two equations:

x%5E3-y%5E3%22%22=%22%2213x-13y

Factor both sides:

%28x-y%29%28x%5E2%2Bxy%2By%5E2%29%22%22=%22%2213%28x-y%29

x and y do not have the same absolute values,
therefore (x-y) does not equal 0 and we may divide
both sides by it.

x%5E2%2Bxy%2By%5E2%22%22=%22%2213

--------------------------

Now we have the two equations:

x%5E2%2Bxy%2By%5E2%22%22=%22%2213
x%5E2-xy%2By%5E2%22%22=%22%2227

Subtract them and get

      2xy%22%22=%22%22-14

      xy+=+-7

Answer: -7

Edwin