How to Calculate the Slope of a Line
Slope is one of the most fundamental concepts in algebra and geometry. It measures the steepness and direction of a line, and it appears in everything from graphing equations to understanding rates of change in data science and physics.
Slope is defined as "rise over run"—how much a line goes up (or down) for every unit it moves to the right.
The Slope Formula
Given two points (x₁, y₁) and (x₂, y₂) on a line:
m = (y₂ − y₁) / (x₂ − x₁)
Where m is the slope.
Step-by-Step Example
Find the slope of the line passing through (2, 3) and (6, 11).
- Identify your points: (x₁, y₁) = (2, 3) and (x₂, y₂) = (6, 11)
- Calculate rise: y₂ − y₁ = 11 − 3 = 8
- Calculate run: x₂ − x₁ = 6 − 2 = 4
- Divide: m = 8 ÷ 4 = 2
The slope is 2, meaning for every 1 unit you move right, the line rises 2 units.
Interpreting Slope
| Slope Value | Meaning |
|---|---|
| m > 0 | Line goes up left to right (positive slope) |
| m < 0 | Line goes down left to right (negative slope) |
| m = 0 | Horizontal line (no rise) |
| Undefined | Vertical line (no run, x₁ = x₂) |
| m = 1 | 45° angle |
| m > 1 | Steeper than 45° |
Real-World Applications
Slope appears in countless real-world scenarios:
- Road grades: A 6% grade means 6 feet of rise for every 100 feet of run (slope = 0.06)
- Roof pitch: A 4/12 pitch means 4 inches of rise for every 12 inches of horizontal run
- Data analysis: In linear regression, slope tells you how much Y changes per unit of X
- Physics: Velocity is the slope of a position-time graph
Special Cases
If x₁ = x₂ (both points have the same x-coordinate), the line is vertical, and the slope is undefined—you can't divide by zero.
If y₁ = y₂, the slope is 0 and the line is perfectly horizontal.
Use our slope calculator to find the slope between any two points instantly.