Math Core

Wheelchair ramps, staircases and roofs all have to be built with the right steepness. In math, the steepness of a line is measured by one number, its slope. Slope also tells you how fast one quantity changes compared with another.

Counting steps on a grid

Think of walking along a line from left to right as climbing a staircase. Each "step" has a rise (how far up or down you go) and a run (how far right you go).

From (0, −1) to (4, 2): right 4, up 3.Open in grapher →

From $(0, -1)$ to $(4, 2)$ you move right 4 and up 3. The slope is $\dfrac{3}{4}$ : the line climbs 3 units for every 4 units you move to the right.

Definition

Slope

The slope of a line is the ratio of its rise to its run. For two points $(x_1, y_1)$ and $(x_2, y_2)$ on the line,

\text{slope} = \frac{\text{rise}}{\text{run}} = \frac{y_2 - y_1}{x_2 - x_1}.

Slope is often written with the letter $m$ .

The slope of a line is the same no matter which two points you use. On the line above, a run of 8 gives a rise of 6, and $\dfrac{6}{8} = \dfrac{3}{4}$ again.

Up, down, flat or straight up

The sign of the slope tells you which way the line goes as you read it left to right.

Four kinds of slope.Open in grapher →

Positive: the line goes uphill.
Negative: the line goes downhill. The rise is negative.
Zero: the line is horizontal. The rise is 0, and $\dfrac{0}{\text{run}} = 0$ .
Undefined: the line is vertical. The run is 0, and dividing by 0 has no answer.

A bigger number (ignoring the sign) means a steeper line. A slope of $-4$ is steeper than a slope of 1.

Using the formula

Counting squares only works when you can see a grid. The formula works for any two points.

Worked example: Slope from two points

Find the slope of the line through $(-3, 6)$ and $(3, 2)$ .

m = \frac{2 - 6}{3 - (-3)} = \frac{-4}{6} = -\frac{2}{3}

The slope is $-\dfrac{2}{3}$ . Sense check: $y$ went down from 6 to 2 while $x$ went up, so the line falls, and a negative slope makes sense.

Common mistake

Keep the points in the same order on the top and the bottom. If $y_2$ comes from the second point, $x_2$ must too. Writing $\dfrac{2 - 6}{-3 - 3}$ mixes the order and gives $\dfrac{-4}{-6} = \dfrac{2}{3}$ , which has the wrong sign.

Slope as a rate of change

In real situations, slope is a rate: how much $y$ changes for each 1 unit of $x$ . Its units are " $y$ -units per $x$ -unit."

Worked example: Earning money

The table shows how much Jordan earns for different numbers of hours worked.

Hours, $x$	2	4	6	8
Dollars, $y$	30	60	90	120

Each time $x$ goes up by 2, $y$ goes up by 30. The slope is $\dfrac{30}{2} = 15$ . Jordan earns $15 per hour.

Worked example: Horizontal or vertical?

Find the slope of the line through $(-2, 5)$ and $(4, 5)$ .
Find the slope of the line through $(1, -3)$ and $(1, 6)$ .

Solutions.

$m = \dfrac{5 - 5}{4 - (-2)} = \dfrac{0}{6} = 0$ . The $y$ -values are equal, so the line is horizontal.
$m = \dfrac{6 - (-3)}{1 - 1} = \dfrac{9}{0}$ , which is undefined. The $x$ -values are equal, so the line is vertical.

Tip

Always simplify the fraction, and put any negative sign out front: $\dfrac{-4}{6} = -\dfrac{2}{3}$ . A slope of $\dfrac{4}{-6}$ means the same thing, but $-\dfrac{2}{3}$ is the standard way to write it.