Math Core

Some hills are gentle and some are steep. Some roads climb and some drop. Slope is a single number that tells you how steep a line is and which way it goes.

Rise over run

Pick two points on a line. To get from the first to the second, you move up or down (the rise) and left or right (the run). Slope compares the two.

From (0, 1) to (3, 7): go right 3, then up 6.Open in grapher →

From $(0, 1)$ to $(3, 7)$ you go right 3 and up 6. The slope is

\text{slope} = \frac{\text{rise}}{\text{run}} = \frac{6}{3} = 2.

A slope of 2 means that for every 1 unit you move right, the line goes up 2 units.

Definition

Slope

The slope $m$ of a line through the points $(x_1, y_1)$ and $(x_2, y_2)$ is

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

Why doesn't it matter which two points you pick? Any two "slope triangles" on the same line are similar: they have the same angles, so their sides are in the same ratio. A triangle with run 1 and rise 2 and a triangle with run 3 and rise 6 give the same slope, 2.

Four kinds of slope

Steeper lines have slopes farther from 0. Lines falling to the right have negative slope.Open in grapher →

Positive slope: the line rises as you move right.
Negative slope: the line falls as you move right.
Zero slope: a horizontal line. The rise is 0.
Undefined slope: a vertical line. The run is 0, and you can't divide by 0.

The bigger the slope's size (ignoring its sign), the steeper the line. A slope of $-3$ is steeper than a slope of 2.

Slope from two points

Worked example: Using the formula

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

m = \frac{2 - 5}{4 - (-2)} = \frac{-3}{6} = -\frac{1}{2}

The slope is $-\dfrac{1}{2}$ : the line drops 1 unit for every 2 units to the right.

y = -x/2 + 4(-2, 5)(4, 2)Open in grapher →

Common mistake

Subtract in the same order on top and bottom. If you start with $y_2$ on top, start with $x_2$ on the bottom. Mixing the order, like $\dfrac{2 - 5}{-2 - 4}$ , flips the sign and gives $\dfrac{1}{2}$ , which is wrong.

Slope from a table

In a table of a linear function, find the change in $y$ and divide by the change in $x$ .

Worked example: A table

$x$	1	3	5	7
$y$	10	16	22	28

Each time $x$ goes up by 2, $y$ goes up by 6. The slope is $\dfrac{6}{2} = 3$ .

Check with the first and last rows: $\dfrac{28 - 10}{7 - 1} = \dfrac{18}{6} = 3$ . ✓

Worked example: Horizontal and vertical lines

Find the slope through $(1, 4)$ and $(6, 4)$ .
Find the slope through $(3, 1)$ and $(3, 5)$ .

Solutions.

$m = \dfrac{4 - 4}{6 - 1} = \dfrac{0}{5} = 0$ . The line is horizontal.
$m = \dfrac{5 - 1}{3 - 3} = \dfrac{4}{0}$ , which is undefined. The line is vertical.

Tip

Before computing, glance at the points. If the line goes up to the right, your slope should be positive. If it goes down, negative. This catches most sign mistakes.