Algebra guide

Point Slope Form

Point-slope form is the fastest way to write the equation of a line when you know a point on the line and its slope. It's a core Algebra I skill that shows up again in precalculus, calculus (tangent lines!), and physics.

The formula

Given a point (x₁, y₁) on a line with slope m:

Deriving it from two points

Suppose you're given two points (x₁, y₁) and (x₂, y₂). First compute the slope:

Then plug either point and the slope into the point-slope formula.

Worked example 1

Find the equation of the line through (2, 3) with slope 4.

Distribute to get slope-intercept form: $y = 4x - 5$.

Worked example 2 (two points)

Find the equation of the line through (1, 2) and (4, 11).

Slope: $m = \frac{11 - 2}{4 - 1} = 3$.

When to use point-slope form

• You're given a point and a slope — write the equation directly.
• You need a tangent line in calculus at x = a: slope is $f'(a)$, point is $(a, f(a))$.
• You want to avoid the extra algebra of solving for b in $y = mx + b$.

Common mistakes

• Sign errors: $y - (-3) = y + 3$.
• Mixing up x₁ and y₁.
• Forgetting to distribute the slope when converting to slope-intercept form.