Decision guideAlgebra IIIntermediate9 min read

Direct or inverse variation: which model matches the relationship?

Direct variation keeps a constant ratio y/x; inverse variation keeps a constant product xy. The data pattern decides the model.

y=kx\qquad y=\frac{k}{x}

Decision guide

A practical comparison that turns a vague choice into a repeatable test.

Updated July 13, 2026

DecisionStart here

If doubling x doubles y, test direct variation. If doubling x halves y, test inverse variation.

Direct variation passes through the origin

In y = kx, the constant k is both the ratio y/x and the slope. When x is zero, y is zero.

A nonzero starting value belongs to a general linear model y = mx + b, not direct variation.

\frac yx=k

In y = k/x, increasing one quantity forces the other down so that xy stays equal to k. The graph has two branches and excludes x = 0.

Travel time for a fixed distance and pressure-volume relationships under controlled conditions are common examples.

xy=k

One pair can determine k for either proposed model. Use another pair to test whether the same ratio or product remains constant.

Real data may only approximate variation, so distinguish an exact algebra exercise from a fitted scientific model.

\frac{y_1}{x_1}=\frac{y_2}{x_2}\quad\text{or}\quad x_1y_1=x_2y_2

Worked exampleIdentify the invariant

\frac{18}{2}=9,\ \frac{12}{3}=4,\ \frac66=1

The ratios are not constant, so it is not direct variation.

2(18)=36,\ 3(12)=36,\ 6(6)=36

The products are constant.

y=\frac{36}{x}

Use k = 36 in the inverse model.

Result

\boxed{y=\frac{36}{x}}

Watch for

Keep