Direct Variation
AlgebraDirect variation describes a relationship where two variables are proportional: y = kx, where k is the constant of variation and the graph always passes through the origin.
Formula
y = kx
Definition
Direct variation means two quantities change together at a constant ratio. When one doubles, the other doubles; when one triples, the other triples. The formula is $y = kx$.
Example
If you earn $\$12$ per hour, your pay $y$ and hours worked $x$ are in direct variation: $y = 12x$. Work $3$ hours: $\$36$. Work $5$ hours: $\$60$. The ratio pay/hours is always $12$.
Key Insight
Direct variation always passes through the origin $(0, 0)$ because when $x = 0$, $y = k \cdot 0 = 0$. If a line does not pass through the origin, it is not direct variation.
Definition
$y$ varies directly with $x$ if $y = kx$ for some non-zero constant $k$, called the constant of variation (or proportionality constant). The graph is a line through the origin with slope $k$. The ratio $y/x = k$ is constant for all $(x, y)$ on the graph. Direct variation is a special case of linear function where $b = 0$.
Example
If $y = 15$ when $x = 3$, find $k$: $k = y/x = 15/3 = 5$. The equation is $y = 5x$. When $x = 7$, $y = 35$.
Key Insight
Direct variation requires both conditions: linear relationship AND passing through the origin. $y = 3x + 2$ is linear but not direct variation (intercept at $2$, not $0$).
Definition
Direct variation $y = kx$ is a linear map from $\mathbb{R}$ to $\mathbb{R}$ with zero constant term, equivalent to a homogeneous linear equation. In abstract terms, it is a morphism of $\mathbb{R}$-modules. More generally, $y$ varies directly as $x^n$ (power variation) means $y = kx^n$, and $y$ varies jointly as $x$ and $z$ means $y = kxz$. These generalize to multivariate direct variation, all sharing the proportionality ratio structure.
Example
Gravitational force $F = Gm_1m_2/r^2$ involves joint variation (directly with $m_1$ and $m_2$) and inverse variation (inversely with $r^2$). Each relationship has its own proportionality constant.
Key Insight
Direct variation is the simplest example of a scaling law. In physics and engineering, scaling laws (dimensional analysis) use the concept of direct variation to predict how quantities scale with fundamental parameters, without solving the full equations.