Prove that the system of equations below has only the solution ( x , y , z ) = ( 1 , 1 , 1 ). { x + y 2 + z 3 = 3 y + z 2 + x 3 = 3 z + x 2 + y 3 = 3

Question

Prove that the system of equations below has only the solution   (  x  ,  y  ,  z  )  =  (  1  ,  1  ,  1  ).      {                            x          +                      y            2                    +                      z            3                    =          3                                      y          +                      z            2                    +                      x            3                    =          3                                      z          +                      x            2                    +                      y            3                    =          3

Colin Moran · Accepted Answer

maybe you could find that   x  ,  y  ,  z are symmetyical ,which means   x  ,  y  ,  z have no difference with each other ,so   x  =  y  =  z.you can solve this original equation by transform to   x  +      x    2    +      x    3    =  3,so   x  =  1,the so the only solution is   (  x  ,  y  ,  z  )  =  (  1  ,  1  ,  1  ).

Celia Lucas · Accepted Answer

Finally, now I can solve my problem with my numbers, thanks!

Prove that the system of equations below has only the solution ( x , y , z )

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