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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:

$y = k x$

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.

$y = k x$

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

$\begin{array}{l} 24 = k \cdot 3 \\ k = 8 \end{array}$

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.

$y = k x$

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

$\begin{array}{l} 30 = k \cdot 6 \\ k = 5 \end{array}$

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

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