How do we solve the system of equations? { x 4 ( 1 4 + 2 x + y x + y ) = 2 y 4 ( 1 4 − 2 x + y x + y ) = 1

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How do we solve the system of equations?      {                                        x            4                                (                                          1                4                                      +                                                            2                                      x                                    +                                      y                                                                    x                  +                  y                                                      )                    =          2                                                  y            4                                (                                          1                4                                      −                                                            2                                      x                                    +                                      y                                                                    x                  +                  y                                                      )                    =          1

frethi38 · Accepted Answer

Divide the first equation by       x    4   and the second by       y    4  . Then adding and subtracting the two resulting equations gives us the new pair of simultaneous equations:      2          x      4        +      1          y      4        =      1    2          2          x      4        −      1          y      4        =  2            2              x            +              y                    x      +      y          .Multiplying these two equations together, we have:      4          x        −      1          y        =            2              x            +              y                    x      +      y          .Clearing denominators yields:  (  4      y    −      x    )  (  x  +  y  )  =  2  x      y    +  y      x  which after some algebra can be reduced to  (  x  +  2  y  )  (  2      y    −      x    )  =  0    .So either   x  =  −  2  y or       x    =  2      y  . If x and   y are positive and real the first is clearly impossible and the second is equivalent to   x  =  4  y. Now, if   x  =  4  y we have            2              y      4        +      1          y      4        =      1    2      ,from the top equation in this answer. So       y    4    =  2  (  1  +      2    ), yielding the solution  x  =  64  (  1  +      2        )    4      y  =  16  (  1  +      2        )    4

mravinjakag · Accepted Answer

According to Maple, there is one real solution,  x  =  1088  +  768      2    ,     y  =  272  +  192      2

How do we solve the system of equations? { <mtable columnalign="left left" rowspacin

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