Manipulating this $\frac{x-y}{z-y}$ to $1+\frac{x-z}{z-y}$

Suppose that we have the expression $\dfrac{x-y}{z-y}$ and we want the numerator to be independent of $y$. Then the only way to do this is to make part of the numerator like the denominator, so that both cancel: $$\frac{x-y}{z-y}=\frac{z-y+x-z}{z-y}=\frac{z-y}{z-y}+\frac{x-z}{z-y}=1+\frac{x-z}{z-y}$$ This is sometimes useful in integration, factorisation and other related topics.

$ \frac{x-y}{z-y} = \frac{z - y + x - z}{z-y} = 1 + \frac{x - z}{z-y} $