Forms of a Linear Equation

y = mx + b gives the slope and the y-intercept at a glance. It is the easiest to graph.

Point-slope form

y − y₁ = m(x − x₁) is what to use when you know a point and the slope, not the intercept.

Standard form

Ax + By = C is convenient for finding both intercepts and for systems.

Parallel and perpendicular

Parallel lines share a slope. Perpendicular slopes multiply to −1, so 2 pairs with −1/2.

Three forms, three purposes

Slope-intercept y = mx + b shows slope and intercept at a glance. Point-slope y − y₁ = m(x − x₁) is easiest when you know a point and the slope. Standard form Ax + By = C suits systems and intercepts.

Converting between them

Every form describes the same line, so you can move between them with ordinary algebra. Expanding point-slope and rearranging gives slope-intercept; rearranging that gives standard form. No information is gained or lost.

Choose the form the data suits

Given two points, use point-slope after computing the slope. Given a rate and a starting value, use slope-intercept directly. Picking the right form first is usually the difference between two lines of work and six.

The line that fits none of them

A vertical line x = 3 has undefined slope and cannot be written as y = mx + b. It can be written in standard form, which is one reason that form is kept. Every general statement about lines needs this exception noted.