Understanding Slope

The slope (also called gradient) of a line measures its steepness — how much the line rises or falls vertically for every unit it moves horizontally. It is often described as "rise over run."

Slope m = (y₂ − y₁) / (x₂ − x₁) = Rise / Run
Line equation: y = mx + b (slope-intercept form)
y-intercept: b = y₁ − m × x₁

Types of Slope

Slope Value	Direction	Meaning	Example
m > 0 (positive)	Rising ↗	Goes up left to right	m = 2: rises 2 for every 1 right
m < 0 (negative)	Falling ↘	Goes down left to right	m = −3: falls 3 for every 1 right
m = 0	Horizontal →	No rise or fall	y = 5 (constant)
m = undefined	Vertical ↕	x does not change	x = 3 (vertical line)

Worked Examples

Example 1 – Basic Slope

Find the slope between (1, 2) and (5, 10).

m = (10−2)/(5−1) = 8/4 = 2
Line equation: b = 2 − 2×1 = 0, so y = 2x

Example 2 – Negative Slope

A road descends from point (0, 100) to point (500, 50). What is the gradient?

m = (50−100)/(500−0) = −50/500 = −0.1
This means a 10% downhill grade — quite steep for a road!

Example 3 – Parallel and Perpendicular Lines

Parallel lines: same slope (m₁ = m₂)
Perpendicular lines: slopes are negative reciprocals
If line 1 has slope m, perpendicular line has slope −1/m
Example: m=3 → perpendicular slope = −1/3

Slope in Real Life

Road gradients: A "10% grade" means the road rises 10 m for every 100 m horizontal distance (slope = 0.10)
Ramps & accessibility: ADA standard requires wheelchair ramps to have maximum slope of 1:12 (m = 1/12 ≈ 0.083)
Roof pitch: A "4/12 pitch" means 4 inches rise for every 12 inches run (m = 1/3)
Economics: The slope of a demand curve tells you how much quantity changes per unit price change
Physics: Slope of a distance-time graph = speed; slope of velocity-time graph = acceleration
Data science: Linear regression finds the best-fit slope through data points

Slope Calculator