# Expressing $\frac{x-y}{w-z}$ as a function of $\frac{x}{w}$ and $\frac{y}{z}$

I have the expression $$\frac{x-y}{w-z}$$ and I would like to rewrite it as a function of two fractions, $$\frac{x}{w}$$ and $$\frac{y}{z}$$. Ideally a linear combination of $$\frac{x}{w}$$ and $$\frac{y}{z}$$ and their powers. If that is not possible, a non-linear expression in $$\frac{x}{w}$$ and $$\frac{y}{z}$$ will also do.

(Apologies in advance for possible tagging this improperly ..)

• It's not possible, let $x/w=a$ and $y/z=b$, then $t=z/w$ is independent of $a,b$, but $\frac{x-y}{w-z}=\frac{a-bt}{1-t}$ depends on $t$.
– Yuz
Mar 20, 2021 at 10:21
• Bascially, you have an equation in terms of four variables, $x, y, w, z$ and you want to rewrite it into two variables, $\frac{x}{w}, \frac{y}{z}$? As a rule of thumb, that will not be possible. Mar 20, 2021 at 10:26

@Babado has written it as a function of $$x/w,\,y/z,\,w,\,z$$. It can't be written as a function of $$x/w,\,y/z$$ alone, because e.g. $$x/w=2,\,y/z=1$$ is compatible with $$\frac{x-y}{w-z}=\frac{2w-z}{w-z}$$ taking any value $$k\notin\{1,\,2\}$$ via $$w/z=(1-k)/(2-k)$$.

You have a function in terms of four variables $$f: (x, y, w, z) \mapsto \frac{x-y}{w-z}$$ and you want to rewrite it in terms of two variables, $$u = \frac{x}{w}, v = \frac{y}{z}$$ so that $$g(u, v) = f(x, y, w, z)$$.

This will only be possible if the function has the property that $$f(tx, sy, tw, sz) = f(x, y, w, z) \quad \forall \ t, s, x, y, w, z \in \mathbb{R}$$ since both $$(x, y, w, z)$$ and $$(tx, sy, tw, sz)$$ map to the same $$u = \frac{x}{w}, v=\frac{y}{z}$$ and thus will have the same corresponding $$g(u, v)$$.

However, \begin{align} &f(tx, sy, tw, sz) = \frac{tx-sy}{tw-sz} = \frac{x-y}{w-z} = \\ \iff& (tx-sy)(w-z) = (tw-sz)(x-y) \\ \iff& t(wx-zx)+s(zy-wy) = t(wx-wy)+s(zy-zx) \\ \iff& t(wx-zx-wx+wy) = s(zy-zx-zy+wy) \\ \iff& t(wy-zx) = s(wy-zx) \\ \iff& wy-zx = 0 \lor t=s \end{align} Which is definitely not always true.

Therefore, there does not exist a function $$g$$ such that $$f(x, y, w, z) = g(u, v)$$.

Thanks to all for sharing. Given that the expansion I seek is not possible in closed form, let me ask further whether it is possible as an approximation.