Show that $xy/z$ is constant given conditions on $xy$ and $y/z$.

Question

If $xy$ is constant whenever $z$ is constant,
and $y/z$ is constant whenever $x$ is constant, then show that $xy/z$ is constant.

My work: Write $xy = a$, $y/z = b$. Then

$$xy^2/z = ab, \ \ \ xz = a/b.$$

Any help?

Remark: This shows up in physics when I'm trying to derive the ideal gas equation from Boyle and Charles' laws ($PV =$ constant and $V/T =$ constant).

Answer 1

My interpretation is that $x, y, z$ are independent, but happens that $xy$ depends only on $z$ and $y/z$ depends only on $x$. Then we write

$$\tag{1} xy = f(z), y/z = g(x)$$ for some functions $f, g$.

A priori, $xy/z$ is a function of $3$ variables and we want to show that it is constant. Since

$$ f(z)/z =\frac{xy}{z} = x g(x),$$

One sees that the expression is independent of $x, y$ and $y, z$. Thus it is independent of $x, y, z$ and thus is a constant.

Answer 2

If $y=z=\frac1x$, then $xy$ and $\frac yz$ are constants. However,$$\frac{xy}z=x,$$which will not be constant unless $x$ itself is constant.

Answer 3

If $xy$ is constant and $z$ is constant, then $\frac{xy}{z}$ must also be a constant.

This is because if $xy = p$ and $z = q$ where $p, q$ are constant, $\frac{xy}{z} = \frac{p}{q}$, and $\frac{p}{q}$ is constant since $p, q$ are constant.

Answer 4

If $\dfrac {xy}{z}$ is constant and $ xy$ is also constant then $z$ is constant.

If in addition $\dfrac{y}{z} $ is also constant, $y,$ and so $(x,y,z)$ are all constants.

Note from Physics about ideal gas Law proportionalities not exactly.

When T is constant pressure P is proportional to 1/V ( Boyle), and when volume V is constant P is proportional to temperature T (Gay-Lussac) and when P is constant V is proportional to T (Charles).

When combined $\dfrac{PV}{T}$ is a new constant.