Rearranging the formula $z = \frac{1-x}{1-yx}$ for $x$ where $0<y<1 ; 0<x<1$

Question

I'm an astronomer and therefore only really use basic algebra, so I understand this is probably a stupidly simple question...

Anyway - the equation $z = \dfrac{1-x}{1-yx}$ has a very bizarre form because of its numerous $\dfrac{1}{x}$ asymptotes. However, where $x$ and $y$ are between $0$ and $1$ (in my case these are ratios - the flux ratio and depth), there is a continuous real solution, and therefore each $x$ & $y$ give a unique $z$ point, and vice-versa each $y$ & $z$ point give a unique $x.$

So it should be possible to rearrange for $x$, such that $x = f(y,z)$. However, I have been unable to find a solution...

Answer 1

$$ \begin{align} z &= \dfrac{1-x}{1-yx} \\ z(1-yx) &= 1-x \\ z - zyx &= 1-x \\ x - zyx &= 1-z \\ x(1-zy) &= 1-z \\ x &= \dfrac{1-z}{1-zy} \end{align} $$

Answer 2

$$z=\frac{1-x}{1-xy} \iff z-zxy=1-x\iff x-zxy=1-z\iff x(1-zy)=1-z \\\iff x=\frac{1-z}{1-zy}$$