This system of equations x y + y z + z x = 3 x 4 + y 4 + z 4 = 3 How to solve this system of equations?

Question

This system of equations                    x        y        +        y        z        +        z        x                            =        3                                                    x          4                +                  y          4                +                  z          4                                    =        3            How to solve this system of equations?

TettetoxDetnhte5 · Accepted Answer

Assuming that you are solving over the reals.  1  =                              x          4                +                  y          4                +                  z          4                    3        4    ≥                              x          2                +                  y          2                +                  z          2                    3        ≥                          |        x        y              |        +              |        y        z              |        +              |        x        z              |            3        ≥                    x        y        +        y        z        +        z        x            3        =  1These inequalities are standard.The first one is power mean applied to fourth powers and second powers.The second one is   (  |  x  |  −  |  y  |      )    2    +  (  |  y  |  −  |  z  |      )    2    +  (  |  z  |  −  |  x  |      )    2    ≥  0The third one follows from the definition of absolute value / triangle inequality.Hence equality must hold throughout, so   |  x  |  =  |  y  |  =  |  z  |  =  1A further check shows that   x  y  +  y  z  +  z  x  =  3 if and only if   x  ,  y  ,  z have the same sign. Thus, we have   x  =  y  =  z  =  1  ,  x  =  y  =  z  =  −  1.Note: As mentioned, the motivation is that the solution set is one where all variables are equal, suggesting that some kind of inequality can be shown to hold, which gives us the equality condition (which is that all variables are equal).

This system of equations <mtable columnalign="right left right left right left right left right

Answered question

Answer & Explanation

New Questions in Algebra I