Proportional Relationship
A proportional relationship exists when two quantities always have a constant ratio, represented by the equation y = kx.
Formula
y = kx \text{ (}k\text{ is the constant of proportionality)}
Definition
A proportional relationship means two quantities always grow or shrink together at the same rate: if you double one, the other doubles too, and the ratio between them is always the same number. Formally, $x$ and $y$ have a proportional relationship if $y = kx$ for some nonzero constant $k$ (the constant of proportionality, or unit rate); the ratio $y/x$ is constant for all pairs, the graph is a straight line through the origin, and doubling $x$ doubles $y$. This is a linear map $f(x) = kx$, a homomorphism of the additive group $\mathbb{R}$ that also satisfies $f(cx) = cf(x)$ (homogeneity of degree $1$); in higher dimensions, proportional relationships generalize to linear maps (matrices), the foundation of linear algebra.
Example
A car traveling at a steady $60$ mph goes $60$ miles in $1$ hour, $120$ miles in $2$ hours, $180$ miles in $3$ hours: miles and hours are proportional, since miles/hours always equals $60$. Checking whether a table represents a proportional relationship: for $(2,6)$, $(5,15)$, $(8,24)$, the ratios $6/2=3$, $15/5=3$, $24/8=3$ are all constant, so $y=3x$ is proportional; but $(2,7)$, $(5,15)$, $(8,24)$ is not, since $7/2=3.5 \neq 15/5=3$. In physics, Ohm's law $V = IR$ is a proportional relationship between voltage and current at constant resistance $R$: doubling $I$ doubles $V$, and the graph of $V$ vs $I$ is a line through the origin with slope $R$.
Key Insight
Proportional relationships always pass through the origin $(0,0)$ on a graph and form a straight line: if someone earns $\$15$/hour, $0$ hours is $\$0$, $1$ hour is $\$15$, $2$ hours is $\$30$, a perfectly straight line through zero. Not all linear functions are proportional, though: $y = kx + b$ with $b \neq 0$ is linear but NOT proportional, because the ratio $y/x$ changes with $x$. The study of proportional relationships is often the first encounter with linear functions and the foundation of algebra; at a deeper level, the axioms of linearity (additivity plus homogeneity) generalize proportional relationships to abstract vector spaces, underpinning all of linear algebra.