If $y$ is proportional to $x$, and $y$ is also proportional to $z$, then how are we able to arrive at the equation: $y = kxz$ ?
My understanding so far of proportionality, which comes from this Wikipedia article, is that two variables are directly proportional if their ratio yields a constant.
Thus, if $y$ is proportional to $x$, then $\frac yx = k$, where $k$ is a constant.
So, if $\frac yx = k*z$, then how are $y$ and $x$ still proportional if $z$ is not a constant?
$\endgroup$1 Answer
$\begingroup$Two variables $\alpha$ and $\beta$ are directly proportional if their ratio is constant, relative to $\alpha$ and $\beta$. That last phrase is missing from the Wiki article, because it is not relevant to the two-variable models discussed there. But it is crucial here.
With that correction, direct proportionality tells us that $$\frac{y}{x}=\lambda_1(z)$$ where $\lambda_1(z)$ is independent of $x$ and $y$, but might depend on $z$. Similarly, $$\frac{y}{z}=\lambda_2(x)$$ where $\lambda_2(x)$ is independent of $y$ and $z$, but might depend on $x$. Dividing the two equations, $$\frac{z}{x}=\frac{\lambda_1(z)}{\lambda_2(x)}$$ or, rearranging, $$\frac{\lambda_1(z)}{z}=\frac{\lambda_2(x)}{x}$$
Each side of the latter equation is constant with respect to the other side, so let their common value be $k$. Then $\lambda_1(z)=kz$, whence the claim.
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