# Direct Variation

Direct variation
describes a simple relationship between two
variables
. We say
$y$
**
varies directly
**
**
with
**
$x$
(or
*
as
*
$x$
, in some textbooks) if:

$y=kx$

for some
constant
$k$
, called the
**
constant of variation
**
or
**
constant of proportionality
**
. (Some textbooks describe direct variation by saying "
$y$
varies directly as
$x$
", "
$y$
varies proportionally as
$x$
", or "
$y$
is directly proportional to
$x$
.")

This means that as $x$ increases, $y$ increases and as $x$ decreases, $y$ decreases—and that the ratio between them always stays the same.

The graph of the direct variation equation is a straight line through the origin.

Direct Variation Equation for $3$ different values of $k$ |

**
Example 1:
**

Given that $y$ varies directly as $x$ , with a constant of variation $k=\frac{1}{3}$ , find $y$ when $x=12$ .

Write the direct variation equation.

$y=\frac{1}{3}x$

Substitute the given $x$ value.

$\begin{array}{l}y=\frac{1}{3}\cdot 12\\ y=4\end{array}$

**
Example 2:
**

Given that $y$ varies directly as $x$ , find the constant of variation if $y=24$ and $x=3$ .

Write the direct variation equation.

$y=kx$

Substitute the given $x$ and $y$ values, and solve for $k$ .

$\begin{array}{l}24=k\cdot 3\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}k=8\end{array}$

**
Example 3:
**

Suppose $y$ varies directly as $x$ , and $y=30$ when $x=6$ . What is the value of $y$ when $x=100$ ?

Write the direct variation equation.

$y=kx$

Substitute the given $x$ and $y$ values, and solve for $k$ .

$\begin{array}{l}30=k\cdot 6\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}k=5\end{array}$

The equation is $y=5x$ . Now substitute $x=100$ and find $y$ .

$\begin{array}{l}y=5\cdot 100\\ y=500\end{array}$