Last updated: 2026-05-19

Definition

The slope of a line measures its steepness — how many units it rises (or falls) for every unit traveled horizontally. Given two points (x₁, y₁) and (x₂, y₂), the slope m is:

$m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{\text{rise}}{\text{run}}$

This ratio is constant for any two points on the same straight line. A slope of 3 means the line climbs 3 units for every 1 unit to the right; a slope of −2 means it drops 2 units per unit of horizontal travel.

Worked example

Take two points: P₁ = (2, 1) and P₂ = (6, 9).

Step 1 — compute rise and run:

$\text{rise} = 9 - 1 = 8, \quad \text{run} = 6 - 2 = 4$

Step 2 — divide:

$m = \frac{8}{4} = 2$

Step 3 — find the angle:

$\theta = \arctan(2) \approx 63.43°$

The line rises steeply: it climbs 2 units vertically for every 1 unit travelled horizontally. That corresponds to an incline of about 63° from the horizontal, steeper than a typical staircase.

Geometric meaning of slope

Slope carries directional information:

The angle from the x-axis (θ) and the slope are linked by θ = arctan(m). For a slope of 1 the angle is exactly 45°; for larger slopes the angle approaches 90°; for negative slopes the angle is negative (below the x-axis).

Why the formula works

The slope formula comes from the definition of a rate of change. A straight line has a constant ratio of vertical to horizontal change everywhere along it. Any two points and the corresponding division of Δy by Δx give that same constant — which is why m fully characterizes a line's tilt. This idea extends into calculus, where the slope of a curve at a point is the derivative — the limit of Δy/Δx as the two points are moved infinitely close together.

Parallel and perpendicular lines

Two lines are parallel when they have the same slope (they never meet). Two lines are perpendicular when their slopes are negative reciprocals:

$m_1 \cdot m_2 = -1 \implies m_2 = -\frac{1}{m_1}$

For example, a line with slope 3 is perpendicular to a line with slope −1/3. This relationship breaks down only at the extremes: a horizontal line (m = 0) is perpendicular to a vertical line (undefined slope).

The vertical-line edge case

When x₁ = x₂, the denominator in the slope formula is zero. Division by zero is undefined, so the slope of a vertical line is undefined — not "infinity." The line still exists and has a direction (straight up), but it cannot be described by y = mx + b. Its equation is simply x = k for a constant k.