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z cannot equal zero......   1/z    in second equation

Multiply both sides of the first equation by $y$ and both sides of the second equation by  to obtain $z$

$\begin{align*} xy+1 &= y \\ yz+1 &= z. \end{align*}$

Substituting $xy+1$ for $y$ in the second equation, we find

$(xy+1)z+1=z,$

which simplifies to

$xyz+z+1=z.$

Subtracting $z+1$ from both sides, we find that $xyz=z-(z+1)=\boxed{-1}.$

x + 1/y  =  1               y +  1/z  =  1

(xy + 1) / y  = 1        (yz  + 1) /  z  = 1

xy +  1 =  y              yz +  1  = z

xy =  y -1                 1 = z - zy

x = (y-1)/y                1 = z ( 1 - y)

                                 z  = 1/(1-y)

So

xyz  =    (y-1) / y  *   y  *  1/(1-y)       =    (y-1)  /  (1-y)  =   -(1-y)  /( 1-y)  =    -1