direct variation

Direct variation describes two quantities that grow together in lockstep: double one and the other doubles too. Buying apples at a fixed price is a perfect example — the total cost varies directly with the number of apples.

We say y varies directly with x if y = kx for some fixed number k, called the constant of variation. Its graph is a straight line through the origin with slope k. The telltale sign is that the ratio y/x stays the same for every pair of values — that constant ratio is k.

Direct variation is exactly the special case of a line with no y-intercept term (b = 0). If the line crosses the y-axis anywhere except the origin, the relationship is linear but not a direct variation, because then doubling x does not simply double y.