Definition

Direct variation describes two quantities that change together at a constant ratio, when one doubles, the other doubles, following the equation $y = kx$ for some non-zero constant $k$, the constant of variation; the graph is always a line through the origin, since $x = 0$ forces $y = k \cdot 0 = 0$, and the ratio $y/x = k$ stays the same everywhere on that line. It is a special case of a linear function with $b = 0$, requiring both a linear relationship and passage through the origin. Formally, $y = kx$ is a linear map from $\mathbb{R}$ to $\mathbb{R}$ with zero constant term, a morphism of $\mathbb{R}$-modules; the idea extends to power variation ($y = kx^n$) and joint variation ($y = kxz$), all sharing the same proportionality structure, and this is the simplest example of a scaling law used throughout physics and engineering.