Direct Variation

Direct variation, also known as direct proportionality or proportional relationship, occurs when two variables,

x

and

y,

have a relationship that forms a linear function passing through the origin.

$y = k x$

k

is the constant of variation. It defines the slope of the line. When

k = 0,

the relationship is not in direct variation. In the example below, the constant of variation is

k = 1.5 .

The constant of variation may be any real number except

0 .

It is worth noting that the quotient of

y

and

x,

when

x \neq = 0,

is the constant of variation.

y = k x \Leftrightarrow \frac{y}{x} = k

Here are some examples.

Examples of Direct Variation
Example	Rule	Comment
The circumference of a circle.	$C = π d$	Here, $d$ is the diameter of the circle and the constant of variation is $π .$
The mass of an object.	$m = ρ V$	Here, $ρ$ is the constant density of the object and $V$ is the volume.
Distance traveled at a constant rate.	$d = r t$	The constant of variation $r$ is the rate and $t$ is the time spent traveling.

Direct variation is closely related to other types of variation.

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