Perpendicular Line Equation: Forms & Steps

Quick answer

y = ax + b with a = -1/m and b = y₀ - a·x₀; or y - y₀ = a(x - x₀).

Introduction

Different classes ask for different forms. The calculator on the home page outputs slope-intercept style for quick graphing.

Start from the perpendicular line formula so the negative reciprocal is already justified before you rearrange symbols.

Point-slope form highlights the given point, which is useful when the problem statement names (x₀, y₀) prominently.

If you need the full numeric path from start to finish, the how to calculate a perpendicular line article matches what most rubrics call complete work.

Choosing a form

Point-slope highlights the given point. Slope-intercept is fastest for graphing. Standard form helps integer-based systems.

Teachers may deduct points if you stop at point-slope when the directions ask for slope-intercept.

Standard form requires integer coefficients when possible, so clear fractions carefully.

All forms describe the same set of points on the plane if your algebra is consistent.

Formula and relationships

• Point-slope
• Slope-intercept
• Standard form Ax + By + C = 0

Rearrange carefully when clearing fractions to satisfy standard-form integer requirements.

The perpendicular slope a is the same in every form; only the arrangement of symbols changes.

Intersection work still uses substitution, regardless of how you wrote the perpendicular line.

Step-by-step guide

1. Find a. Negative reciprocal of the base slope.
2. Write point-slope. Substitute (x₀, y₀).
3. Expand to slope-intercept. Distribute and combine terms if required.
4. Convert to standard form. Move terms and clear fractions when needed.
5. Check with substitution. Plug the given point into your final equation.

Worked examples

Through (4, -1) perpendicular to a line with slope 2: a = -1/2.

Point-slope: y + 1 = -0.5(x - 4). Slope-intercept: y = -0.5x + 1.

Standard form might read x + 2y - 2 = 0 after clearing fractions.