Slope: how fast a line climbs

A ramp, a staircase, a phone plan that charges per gigabyte — each has a rate: how much one thing changes when another changes. On a graph, that rate has a shape. It's the steepness of a line, and algebra calls it slope.

Rise over run

Pick any two points on a line. The rise is how far you go up (or down) between them; the run is how far you go across. Slope is their ratio:

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

The picture shows the punchline: draw a small triangle or a big one, anywhere on the line — the ratio never changes. A line is exactly the curve whose rate is constant.

Computing it

From the points (1, 3) and (5, 11):

$m = \frac{11 - 3}{5 - 1} = \frac{8}{4} = 2$

Up 2 for every 1 across. Order doesn't matter — subtract the other way and both signs flip, leaving the same slope. Just don't mix orders between top and bottom.

The four personalities

A vertical line has no slope — its run is zero, and rise ÷ 0 is undefined. That's not a technicality: "infinite steepness" isn't a rate you can charge per gigabyte.

Why this matters

Slope is the single most reused idea in this course: parallel lines share it, systems cross because theirs differ, and y = mx + b (next lesson) is built around it. Later, calculus begins by asking what slope means for curves.