Inverse Variation

In science and mathematics, we often look for relationships between two quantities. Sometimes when one quantity increases, the other also increases — that's direct variation. But sometimes the opposite happens: as one quantity increases, the other decreases in a very specific, predictable way. This is called inverse variation.

The key signature of inverse variation is simple: multiply the two values together and you always get the same number — the constant of variation, usually written as $k$.

$x\times y=k$

or equivalently:

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

The general equation for inverse variation is:

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

where:

• $y$ is the dependent variable
• $x$ is the independent variable
• $k$ is the constant of variation (always non-zero)

This can be rearranged to: $xy=k$

This rearranged form is extremely useful — it tells us that any pair of corresponding values $(x_1,y_1)$ and $(x_2,y_2)$ in an inverse variation will satisfy:

$x_1{y}_1=x_2{y}_2=k$