Direct and inverse proportionality

Question

Let $x$ is directly proportional to $y$ and inversely proportional to $z$.

From above condition we have that: $x=\alpha y$ and $x=\dfrac{\beta}{z}$.

Does it follow that $x=k\dfrac{y}{z}$ where $k$ is some constant?

If it is true could anyone explain how it follows?

Related: https://math.stackexchange.com/questions/2753852/joint-proportionality — Henry, Apr 30 '18 at 17:02
Does this answer your question? Does $x$ being directly proportional to $y$ and $z$ separately imply that $x$ is directly proportional to $yz?$ $\quad$ If there is no other relationship between $y$ and $z,$ then the $\alpha$ that you write cannot be a real number; if $x=f(y,z),$ then $\alpha=g(z).$ — ryang, Jun 25 '23 at 15:15

Agile_Eagle · Answer 1 · 2017-08-12T12:15:20.437

0

As you said,

$$ x=\alpha y $$

and

$$ x=\frac{\beta}{z} $$

Multiplying the two equations we get,

$$ x^2 = \alpha \beta\frac{y}{z} $$

So, the correct relation is,

$$ x=k_1 \sqrt{\frac{y}{z}} $$

Now, let us suppose that $x=k_2\frac{y}{z}$ is also true.

Then,

$${\frac{y}{z}} = k_3$$

which is false since you placed no constraints on $y$ and $z$.

edited Aug 12 '17 at 12:15

answered Aug 12 '17 at 10:39

Agile_Eagle

1 Answers1