Slope of a Line

Rise over run — the steepness of a line between two points.

The idea

The slope of a line measures its steepness: how many units of $y$ the line gains for each unit of $x$ moved to the right. One number is enough because a line climbs at the same rate everywhere along its length.

Definition (Slope).

Take any two points $(x_1, y_1)$ and $(x_2, y_2)$ on the line. Travelling from the first to the second, the rise is the change in $y$ and the run is the change in $x$; the slope is their quotient, $m = \frac{y_2 - y_1}{x_2 - x_1}.$

A positive $m$ means the line climbs, a negative $m$ means it falls, and the farther $m$ is from zero the steeper the line.

Subtract the coordinates in the same order in the numerator and the denominator; reversing only one of them flips the sign of the answer. Finally, two points stacked vertically give a run of zero, and the formula would divide by zero: a vertical line has no slope at all.

Ways to work on it

• Walkthrough. Define slope and handle horizontal and vertical lines.
• Practice. Slope between two points with a clean integer answer.
• Hardest. Fractional slopes and recovering a missing coordinate.

Not sure where to start? Take the ten-question placement test.