Can one combine proportions like so?

If $$y \propto x$$ and $$y \propto \frac1z$$ does this imply $$y \propto \frac{x}{z}$$ is true? I've tried a few cases and it has worked every time, however I can't find any information on the internet and I'm not sure how to derive a proof.

• Sometime, they called joint variation. – Ng Chung Tak Apr 25 at 19:40
• Thank you, just what i was looking for. I have no idea why i couldn't find this. – user668526 Apr 25 at 19:42

$$y\propto x$$ means $$y=k_1x$$, for some $$k_1$$ when $$z$$ is constant. $$y\propto\frac 1z$$ when $$x$$ is constant. From second proportionality, we can rewrite $$k_1x\propto 1/z$$ when $$x$$ is const. As, $$x$$ is const. we have $$k_1\propto k_2/z$$, we can write it like $$k_1=k_3/z$$ or, $$y=\frac{k_3x}{z}$$ which means $$y\propto\frac xz$$
No, it's not true in general, because the variables $$x$$ and $$z$$ are often not independent. For example, suppose $$x$$ and $$z$$ are constrained so that $$xz = 1$$. To be even more explicit, suppose the set of attainable triples $$(x,y,z)$$ is $$\{(t, t, 1/t) : t > 0\}$$. Then certainly $$y \propto x$$ and $$y \propto 1/z$$, but $$x/z = y^2$$, so $$y \propto \sqrt{x/z}$$.