# Joint Variation

Joint variation describes a situation where one variable depends on two (or more) other variables, and varies directly as each of them when the others are held constant. We say $z$ varies jointly as $x$ and $y$ if

$z=kxy$

for some constant $k$.

Example:

If $z$ is jointly proportional to $x$ and $y$ and $z=6$, when $x=3$ and $y=4$, find $z$ when $x=7$ and $y=4$.

Find $k$:

$6=3\left(4\right)k$

$k=\frac{1}{2}$

Then, find $z$ when $x=7$ and $y=2$.

$\begin{array}{l}z=\frac{1}{2}\left(7\right)\left(2\right)\\ z=7\end{array}$