# Combined Variation

Combined variation describes a situation where a variable depends on two (or more) other variables, and varies directly with some of them and varies inversely with others (when the rest of the variables are held constant). For example, if $z$ varies directly as $x$ and inversely as $y$ , we have the following combined variation equation:

$z=k\left(\frac{x}{y}\right)$

Example:

If $x$ varies directly as $y$ and inversely as $z$ ,
and $x=10$ when $y=5$ and $z=3$ , for what value of $z$ will $x=3$ and $y=4$ ?

$\begin{array}{l}x=k\cdot \frac{y}{z}\\ 10=k\cdot \frac{5}{3}\\ k=6\end{array}$

To find $z$ , when $x=3$ and $y=4$

$\begin{array}{l}3=6\cdot \frac{4}{z}\\ 3z=24\\ z=8\end{array}$