Equations of Linear Functions

Point-slope to slope-intercept

Point-slope form is $y - y_1 = m(x - x_1)$, where $m$ is the slope and $(x_1, y_1)$ is the known point. Plug those in, then distribute and solve for $y$ to reach slope-intercept form.

For slope $\dfrac{1}{2}$ through $(4, -1)$: $y + 1 = \dfrac{1}{2}(x - 4)$, which simplifies to $y = \dfrac{1}{2}x - 3$.

Parallel and perpendicular slopes

Parallel lines have equal slopes. Perpendicular lines have slopes that are negative reciprocals — flip and change the sign, so $2$ pairs with $-\dfrac{1}{2}$.

To write a parallel or perpendicular line through a point, choose the right slope by this rule, then use point-slope form as usual.

Below, a line of slope $2$ (blue) and a line of slope $-\dfrac{1}{2}$ (red) cross at a right angle — negative-reciprocal slopes always do.

Worked examples

Try one yourself

Common questions