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.
LaTeX article Updated July 13, 2026
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.
Inverse variation keeps a product constant
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.
Test more than one data pair
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.
Worked example
Common mistakes
- Calling every line a direct variation.
- Checking only one data pair.
- Forgetting that inverse variation excludes x = 0.
Keep these ideas
- Direct variation preserves a ratio.
- Inverse variation preserves a product.
- Use multiple data pairs to test the model.