How many solutions are there to this equation? x 2 − y 2 = z y 2 − z 2 = x z 2 − x 2 = y

Question

How many solutions are there to this equation?                              x          2                −                  y          2                                    =        z                                      y          2                −                  z          2                                    =        x                                      z          2                −                  x          2                                    =        y

klemmepk · Accepted Answer

We have                              x          2                −                  y          2                                    =        z                                      y          2                −                  z          2                                    =        x                                      z          2                −                  x          2                                    =        y            Adding the first two equations, we get that      x    2    −      z    2    =  z  +  x    ⟹    z  =  −  x   or   x  −  z  =  1Similarly, we get that  x  =  −  y   or   y  −  x  =  1  y  =  −  z   or   z  −  y  =  1Hence, there are   8 possible choices of getting   3 equations. But the choice  x  −  z  =  1   and   y  −  x  =  1   and   z  −  y  =  1gives no solution. The rest of the seen choices give the following solutions  (  0  ,  0  ,  0  )  ;  (  −  1  ,  0  ,  1  )  ;  (  0  ,  1  ,  −  1  )  ;  (  1  ,  −  1  ,  0  )

Roland Waters · Accepted Answer

Sum both sides and get   x  +  y  +  z  =  0, so you&#039;re on a plane.Further hint:  z  =  (      x    2    −      y    2    )  =  (  x  −  y  )  (  x  +  y  )  =  −  z  (  x  −  y  )    ⟹    …

How many solutions are there to this equation? <mtable columnalign="right left right left right

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