Suppose that x , y , z are three positive numbers that satisfy the equation x y z = 1 , x + 1 z = 5 and y + 1 x = 29. Then z + 1 y = m n , where m and n are coprime. Find m + n + 1

Question

Suppose that   x  ,  y  ,  z are three positive numbers that satisfy the equation   x  y  z  =  1  ,  x  +      1    z    =  5 and   y  +      1    x    =  29. Then   z  +      1    y    =      m    n  , where m and n are coprime. Find   m  +  n  +  1

grcalia1 · Accepted Answer

Just substitute the values of y and z in   x  y  z  =  1            x      (      29      x      −      1      )              x      (      5      −      x      )        =  1  29  x  −  1  =  5  −  x  x  =      1    5    y  =            29              /            5      −      1              1              /            5        =  24  z  =      1          5      −      1              /            5        =      5    24    z  +      1    y    =      5    24    +      1    24    =      1    4  Thus,  m  +  n  +  1  =  1  +  4  +  1  =  6

Garrett Black · Accepted Answer

Another solution: we have       1    x    =  y  z,       1    y    =  x  z, and       1    z    =  x  y. So                    x        +        x        y                            =        5                            y        +        y        z                            =        29                            z        +        x        z                            =        Q            where Q is the quantity we&#039;re trying to determine.Multiplying these three equations together gives   145  Q  =  x  y  z  (  x  +  1  )  (  y  +  1  )  (  z  +  1  ) and hence   145  Q  =  (  x  +  1  )  (  y  +  1  )  (  z  +  1  )Expanding, we have                    (        x        +        1        )        (        y        +        1        )        (        z        +        1        )                            =        x        y        z        +        x        y        +        y        z        +        x        z        +        x        +        y        +        z        +        1                                          =        2        +        x        y        +        y        z        +        x        z        +        x        +        y        +        z                                          =        2        +        5        +        29        +        Q                                          =        36        +        Q            and so   145  Q  =  36  +  Q, from which it follows that   Q  =      1    4

Suppose that x , y , z are three positive numbers that satisfy the equation x y z

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