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Concept

Inverse Variation

Inverse variation, or inverse proportionality, occurs when two variables, $x$ and $y,$ have a relationship such that their product is constant. This relationship is often written with one of the variables isolated on the left-hand side.

$k=xy\quad \Leftrightarrow \quad y=\dfrac{k}{x}$

The constant, $k,$ is called the constant of variation or the proportionality constant.

The constant of variation may be any real number except $0.$

Examples of Inverse Variation
Example Rule Comment
The gas pressure in a sealed container if the container's volume is changed. $p=\dfrac{k}{V}$ $k$ depends on the amount of gas there is in the container and the temperature.
The time, $t,$ it takes to travel a given distance at various speeds, $s.$ $t=\dfrac{d}{s}$ The constant of variation, $d,$ is the distance.

Inverse variation is closely related to other types of variation.