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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 ..)

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    $\begingroup$ 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 $t$. $\endgroup$
    – Yuz
    Mar 20, 2021 at 10:21
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    $\begingroup$ 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. $\endgroup$
    – Poseidaan
    Mar 20, 2021 at 10:26

3 Answers 3

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@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 $w/z=(1-k)/(2-k)$.

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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 $f(x, y, w, z) = g(u, v)$.

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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.

[Moderators please advise whether I need to post under a new thread]

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