Suppose that x , y and z are three integers (positive,negative or zero) such that w

Case Nixon 2022-05-23 Answered
Suppose that x, y and z are three integers (positive,negative or zero) such that we get the following relationships simultaneously
1. x + y = 1 z
and
2. x 3 + y 3 = 1 z 2
Find all such x, y and z.
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Answers (1)

Norah Baxter
Answered 2022-05-24 Author has 10 answers
If z = 1, then y = x and all triples ( x , x , 1 ) are solutions. So let's suppose z 1. Then we can divide
(3) x 2 x y + y 2 = x 3 + y 3 x + y = 1 z 2 1 z = 1 + z .
Adding ( ( 3 ) to ( 1 ) yields
x 2 x y + y 2 + x + y = 2.
Multiplying by 2 and adding 2 to both sides yields
6 = 2 x 2 2 x y + 2 y 2 + 2 x + 2 y + 2 = ( x 2 2 x y + y 2 ) + ( x 2 + 2 x + 1 ) + ( y 2 + 2 y + 1 ) = ( x y ) 2 + ( x + 1 ) 2 + ( y + 1 ) 2 .
There aren't many combinations of x and y left to check.
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