Lesson 22 — Equation of a Line

Forms of the line equation

Slope-intercept form

$y = mx + n$, where $m$ is the slope and $n$ is the y-intercept.

General form

$Ax + By + C = 0$, with $(A, B) \neq (0, 0)$. For non-vertical lines ($B \neq 0$): $m = -A/B$ and $n = -C/B$.

Point-slope form

Line through $(x_0, y_0)$ with slope $m$: $y - y_0 = m(x - x_0)$

Parametric form

Line with direction vector $\vec u = (a, b)$ through $(x_0, y_0)$: $\begin{cases} x = x_0 + at \\ y = y_0 + bt \end{cases}, \quad t \in \mathbb{R}$

Equation through two points

Line through $(x_1, y_1)$ and $(x_2, y_2)$: $\frac{y - y_1}{x - x_1} = \frac{y_2 - y_1}{x_2 - x_1}$

Slope from two points

$m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{\Delta y}{\Delta x} = \tan\alpha$

where $\alpha$ is the angle of the line with the $x$-axis.

Table of the 5 forms