ABC26GN0977 · Simplification

$$\begin{aligned} y+\frac{1}{z}=1 & \Rightarrow y z+1=z \Rightarrow y z-z=-1 \\ & \Rightarrow z(y-1)=-1 \Rightarrow z=\frac{1}{(1-y)} \\ x+\frac{1}{y}=1 & \Rightarrow x y+1=y \Rightarrow x y-y=-1 \\ & \Rightarrow y(x-1)=-1 \Rightarrow y=\frac{1}{(1-x)} . \\ \therefore \quad x y z= & x\left(\frac{1}{1-x}\right)\left(\frac{1}{1-y}\right)=x\left(\frac{1}{1-x}\right)\left(\frac{1}{1-\frac{1}{1-x}}\right) \\ = & x\left(\frac{1}{1-x}\right)\left(\frac{1-x}{-x}\right)=-1 . \end{aligned}$$