Slope Formula
Calculate the slope of a line between two points using m = (y2-y1)/(x2-x1).
Covers positive, negative, zero, and undefined slopes with real-world examples.
The Formula
m = (y₂ - y₁) / (x₂ - x₁)
The slope formula measures how steep a line is.
It tells you how much y changes for every one-unit change in x.
Variables
| Symbol | Meaning |
|---|---|
| m | The slope of the line (rise over run) |
| (x₁, y₁) | Coordinates of the first point |
| (x₂, y₂) | Coordinates of the second point |
Understanding Slope Values
- Positive slope → line goes uphill (left to right)
- Negative slope → line goes downhill (left to right)
- Zero slope → horizontal line
- Undefined slope → vertical line (x₂ - x₁ = 0)
Example 1
Find the slope between (1, 3) and (5, 11)
x₁ = 1, y₁ = 3, x₂ = 5, y₂ = 11
m = (11 - 3) / (5 - 1)
m = 8 / 4
m = 2 (the line rises 2 units for every 1 unit to the right)
Example 2
Find the slope between (2, 9) and (6, 1)
x₁ = 2, y₁ = 9, x₂ = 6, y₂ = 1
m = (1 - 9) / (6 - 2)
m = -8 / 4
m = -2 (the line falls 2 units for every 1 unit to the right)
When to Use It
Use the slope formula when:
- Determining the steepness or direction of a line
- Writing the equation of a line (slope-intercept form: y = mx + b)
- Checking if two lines are parallel (same slope) or perpendicular (negative reciprocal slopes)
- Calculating rate of change in real-world problems (speed, cost per unit, etc.)
Key Notes
- Formula: m = (y₂ − y₁) / (x₂ − x₁) = rise / run: The slope measures the steepness and direction of a line. Positive slope: rises left to right. Negative slope: falls left to right. The same slope results regardless of which point is labeled 1 or 2.
- Parallel and perpendicular lines: Parallel lines have equal slopes (m₁ = m₂). Perpendicular lines have slopes that are negative reciprocals: m₁ × m₂ = −1, or m₂ = −1/m₁. A horizontal line (slope 0) is perpendicular to a vertical line (undefined slope).
- Undefined vs zero slope: A vertical line (x = constant) has undefined slope — the denominator (x₂ − x₁) is zero. A horizontal line (y = constant) has slope = 0 — a defined value. These are fundamentally different cases, not interchangeable.
- Slope as rate of change: Slope is the rate of change of y with respect to x. In physics, position vs time slope = velocity; velocity vs time slope = acceleration. In economics, total cost vs quantity slope = marginal cost. In calculus, slope generalizes to the derivative.
- Applications: Slope calculations are used in road grade design (grade = rise/run as a percentage), roof pitch, linear regression (best-fit slope), physics kinematics graphs, engineering tolerances for drainage, and economic analysis of linear cost/revenue models.