CEMC Fermat

Since the problem asks us to find the value of $\frac{x-z}{y-z}$, then this value must be the same no matter what $x, y$ and $z$ we choose that satisfy $\frac{x-y}{z-y}=-10$. Thus, if we can find numbers $x, y$ and $z$ that give $\frac{x-y}{z-y}=-10$, then these numbers must give the desired value for $\frac{x-z}{y-z}$. If $x=10, y=0$ and $z=-1$, then $\frac{x-y}{z-y}=-10$. In this case, $\frac{x-z}{y-z}=\frac{10-(-1)}{0-(-1)}=\frac{11}{1}=11$.