Understand Slope of a Line

When you graph linear equations, you may notice that some lines tilt up as they go from left to right and some tilt down. Some lines are steep and some are flatter. What determines whether a line tilts up or down, or how steep or flat it is?

In mathematics, the “tilt” of a line is called the slope of the line. The slope of a line is the ratio of the vertical change to the horizontal change between any two points on the line.

Use $m = \tfrac{\text{rise}}{\text{run}}$ to find the slope of a line from its graph

To find the slope of a line from its graph, we locate two points on the line whose coordinates are integers. Starting with the point on the left, we sketch a right triangle, going from the first point to the second, so we can count the rise and the run.

Example. Find the slope of the line shown.

We locate two points with integer coordinates, $(0, -3)$ and $(5, 1)$. Starting at the point on the left, $(0, -3)$, we sketch a right triangle to $(5, 1)$. The rise is $4$ and the run is $5$, so:

The slope of the line is $\tfrac{4}{5}$. This means that $y$ increases $4$ units as $x$ increases $5$ units.

It does not matter which of the two points you start from, or which point you call “first” — the slope of the line is always the same.

Find the slope of horizontal and vertical lines

Horizontal and vertical lines have equations with just one variable — $y = b$ for a horizontal line and $x = a$ for a vertical line. What is their slope?

For a horizontal line, any two points on it have the same $y$-coordinate, so the rise between them is always $0$. Since $m = \tfrac{\text{rise}}{\text{run}} = \tfrac{0}{\text{run}} = 0$, every horizontal line has slope $0$.

For a vertical line, any two points on it have the same $x$-coordinate, so the run between them is always $0$. Since division by $0$ is undefined, the slope of every vertical line is undefined.

Example. Find the slope of each line: (a) $x = 8$ (b) $y = -5$.

(a) $x = 8$ is a vertical line. Its slope is undefined.

(b) $y = -5$ is a horizontal line. It has slope $0$.

Use the slope formula to find the slope of a line between two points

Sometimes we need to find the slope of a line between two points without a graph to count the rise and run. To do this algebraically, we use subscript notation: $(x_1, y_1)$ ("$x$ sub $1$, $y$ sub $1$") names the first point and $(x_2, y_2)$ ("$x$ sub $2$, $y$ sub $2$") names the second.

The rise between two points is the difference in their $y$-coordinates, $y_2 - y_1$, and the run is the difference in their $x$-coordinates, $x_2 - x_1$. Substituting these into $m = \tfrac{\text{rise}}{\text{run}}$ gives the slope formula.

Example. Use the slope formula to find the slope of the line through the points $(1, 2)$ and $(4, 5)$.

We call $(1, 2)$ point #1 and $(4, 5)$ point #2, so $(x_1, y_1) = (1, 2)$ and $(x_2, y_2) = (4, 5)$. Substituting into the slope formula:

It does not matter which point you call point #1 and which you call point #2 — the slope will be the same either way.

Example. Use the slope formula to find the slope of the line through the points $(-2, -3)$ and $(-7, 4)$.

Graph a line given a point and the slope

We have graphed lines by plotting points, by using intercepts, and by recognizing horizontal and vertical lines. Another method, the point-slope method, graphs a line from one known point and the slope: we plot the point, then use the definition of slope to find and mark a second point.

Example. Graph the line passing through the point $(1, -1)$ whose slope is $m = \tfrac{3}{4}$.

We plot $(1, -1)$. The slope $m = \tfrac{3}{4}$ gives rise $= 3$ and run $= 4$. Starting at $(1, -1)$, we count up $3$ and right $4$ to mark the second point, $(5, 2)$, then connect the two points with a line.

Solve slope applications

Slope has many applications in the real world — the pitch of a roof, the grade of a highway, and the drop of a pipe are all slopes.

Example. The pitch of a building’s roof is the slope of the roof. A roof rises $9$ feet over a run of $18$ feet. What is the slope of the roof?

The roof rises $1$ foot for every $2$ feet of horizontal run.

Example. Sewage pipes must slope down $\tfrac{1}{4}$ inch per foot in order to drain properly. What is the required slope?

Since $1$ foot is $12$ inches, and the pipe drops (a negative rise) of $\tfrac{1}{4}$ inch:

The pipe drops $1$ inch for every $48$ inches of horizontal run.

Key terms

slope of a line — the ratio of the rise (vertical change) to the run (horizontal change) between two points on the line, $m = \tfrac{\text{rise}}{\text{run}} = \tfrac{y_2 - y_1}{x_2 - x_1}$. slope formula — the algebraic formula for computing slope from two named points, $(x_1, y_1)$ and $(x_2, y_2)$. point-slope method — graphing a line by plotting one known point, then using the slope to count out and mark a second point.

This section is adapted from Elementary Algebra 2e, Section 4.4: Understand Slope of a Line by Lynn Marecek, MaryAnne Anthony-Smith, and Andrea Honeycutt Mathis, © OpenStax, licensed under CC BY-NC-SA 4.0. Access the original for free at openstax.org. Changes: recreated the geoboard and coordinate-plane figures as accessible inline graphics and condensed the geoboard-modeling activity into a single worked introduction of rise and run; omitted the Be Prepared quiz, Media links, and Section Exercises; and converted the practice problems (“Try Its”) into interactive exercises with instant feedback.