A nonlinear system of equations { <mtable columnalign="right center left" rowspacing

Dayami Rose

Dayami Rose

Answered question

2022-06-11

A nonlinear system of equations
{ y 3 9 x 2 + 27 x 27 = 0 z 3 9 y 2 + 27 y 27 = 0 x 3 9 z 2 + 27 z 27 = 0

Answer & Explanation

livin4him777lf

livin4him777lf

Beginner2022-06-12Added 14 answers

For integer solutions:
You can rewrite the equations using
y 3 9 x 2 + 27 x 27 = y 3 x 3 + ( x 3 ) 3
and similarly for the two other equations. Adding the three equations then yields
( x 3 ) 3 + ( y 3 ) 3 + ( z 3 ) 3 = 0.
It is long known that in all integer solutions of the equation a 3 + b 3 = c 3 , one of the integers must be 0. Our equation is (modulo rearranging) exactly that, so we must have x = 3 or y = 3 or z = 3.
Suppose x = 3. Then the first equation reduces to y 3 x 3 = 0, hence y 3 = 27, hence y = 3. In the same way, it follows that z = 3. Starting from y = 3 or z = 3 yields the same. So ( 3 , 3 , 3 ) is the only integer solution to the system.
Now, for any real solution, we still have by adding the equations
( x 3 ) 3 + ( y 3 ) 3 + ( z 3 ) 3 = 0 ,
so if one of x , y , z were different from 3, one of the three at least would be larger than 3. Without loss of generality, let x = 3 + δ > 3. Then, from the first equation, we obtain
y 3 = x 3 ( x 3 ) 3 = ( 3 + δ ) 3 δ 3 = 3 3 + 27 δ + 9 δ 2 > 3 3 ,
so also y > 3. The same reasoning for the second equation then yields z > 3, and so
( x 3 ) 3 + ( y 3 ) 3 + ( z 3 ) 3 > 0
contradicting the assumption that ( x , y , z ) is a solution of the system.
Hence ( 3 , 3 , 3 ) is the only real solution of the system.

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