Definition
A linear equation in two variables expresses one quantity as a constant multiple of another, plus an offset. In the canonical slope-intercept form y = mx + b, the coefficient m is the slope — the change in y for each one-unit change in x — and b is the intercept, the value of y when x equals zero. The graph of the equation is always a straight line, which is what gives the form its name.
Linear equations are the simplest non-trivial functional relationships and the first model an analyst reaches for when two quantities are suspected to vary together at a roughly constant rate. They are also the building block of countless higher-level tools, from regression to optimization to differential equations.
Why it matters
How it works
The slope of a linear equation is computed as the rise over the run between any two points on the line: subtract the y-values, subtract the x-values, divide. Because the line is straight, this ratio is the same no matter which pair of points you pick — that constancy is the defining property of linearity. The intercept is then found by substituting any known point on the line into the equation and solving for b. Together, the slope and intercept fully determine the line and let you predict the y-value for any x-value you choose.
A linear equation extends naturally to more variables. With two predictors, the equation becomes y = m1 * x1 + m2 * x2 + b, and the graph is a tilted plane rather than a line. With more predictors, the picture moves into higher-dimensional space and ceases to be visualizable, but the equation retains its essential form: each predictor contributes its own constant rate to the response. This generalization is what powers multiple linear regression and a wide span of statistical and engineering models.