System of equations with multiplicationI want to find all x , y , z ∈ R such that ( x + 1 ) y z = 12 , ( y + 1 ) z x = 4 , ( z + 1 ) x y = 4.I can multiply all three equations to get ( x + 1 ) ( y + 1 ) ( z + 1 ) x 2 y 2 z 2 = 192I can divide the first equation by the second to get ( x + 1 ) y ( y + 1 ) x = 4, which simplifies to 3 x y + 4 x − y = 0None of these seems to help. How can I solve the equations?

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System of equations with multiplicationI want to find all   x  ,  y  ,  z  ∈  R such that  (  x  +  1  )  y  z  =  12  ,  (  y  +  1  )  z  x  =  4  ,  (  z  +  1  )  x  y  =  4.I can multiply all three equations to get  (  x  +  1  )  (  y  +  1  )  (  z  +  1  )      x    2        y    2        z    2    =  192I can divide the first equation by the second to get                    (        x        +        1        )        y                    (        y        +        1        )        x              =  4, which simplifies to   3  x  y  +  4  x  −  y  =  0None of these seems to help. How can I solve the equations?

Marco Meyer · Accepted Answer

Your equations are                    I                                    x        y        z        +        y        z                    =        12                            I        I                                    x        y        z        +        x        z                    =        4                            I        I        I                                    x        y        z        +        x        y                    =        4            Well, now:                    I        −        I        I        :                                            (        x        −        y        )        z        =        8                            I        −        I        I        I        :                                            (        x        −        z        )        y        =        8                            I        I        −        I        I        I        :                                            (        z        −        y        )        x        =        0            Since     x  ≠  0  , we get     y  =  z   and thus     x  z  =  x  y  =      y    2    +  8   ...take it from here.

System of equations with multiplication I want to find all x , y , z ∈<!-

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