Slope of a Line

Find the 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 lines tilt down. Some lines are very steep and some lines are flatter.

In mathematics, the measure of the steepness of a line is called the slope of the line. The concept of slope has many applications in the real world. In construction, the pitch of a roof, the slant of the plumbing pipes, and the steepness of the stairs are all applications of slope, and as you ski or jog down a hill, you definitely experience slope.

We can assign a numerical value to the slope of a line by finding the ratio of the rise and run. The rise is the amount the vertical distance changes while the run measures the horizontal change. Slope is a rate of change.

To find the slope of a line, we locate two points on the line whose coordinates are integers. Then we sketch a right triangle where the two points are vertices and one side is horizontal and one side is vertical. We measure the distance along the vertical and horizontal sides of the triangle. The vertical distance is called the rise and the horizontal distance is called the run.

Example. Find the slope of the line shown.

Locate two points on the graph whose coordinates are integers: $(0,5)$ and $(3,3)$. Starting at $(0,5)$, sketch a right triangle to $(3,3)$. Count the rise—since it goes down, it is negative. The rise is $-2$. Count the run. The run is $3$. Use the slope formula and substitute the values:

The slope of the line is $-\tfrac{2}{3}$. So $y$ decreases by $2$ units as $x$ increases by $3$ units.

How do we find the slope of horizontal and vertical lines? For the horizontal line $y=4$, the rise is $0$ and the run is $3$, so $m=\tfrac{0}{3}=0$. For the vertical line $x=3$, the rise is $2$ and the run is $0$. Its slope is undefined since division by zero 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 $0$.

Sometimes we’ll need to find the slope of a line between two points when we don’t have a graph to count out the rise and the run. We could plot the points on grid paper, then count out the rise and the run, but there is a way to find the slope without graphing.

We use $(x_1,y_1)$ to identify the first point and $(x_2,y_2)$ to identify the second point. The rise can be found by subtracting the $y$-coordinates, and the run can be found by subtracting the $x$-coordinates.

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

We’ll call $(-2,-3)$ point #1 and $(-7,4)$ point #2. Use the slope formula, substitute the values, and simplify:

Graph a line given a point and the slope

Up to now, in this chapter, we have graphed lines by plotting points, by using intercepts, and by recognizing horizontal and vertical lines. We can also graph a line when we know one point and the slope of the line. We will start by plotting the point and then use the definition of slope to draw the graph of the line.

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

Plot $(1,-1)$. Identify the rise and run: $m=\tfrac{3}{4}$, so rise $=3$ and run $=4$. Start at $(1,-1)$ and count up $3$ units and right $4$ units. Connect the two points with a line.

You can check your work by finding a third point. Since the slope is $m=\tfrac{3}{4}$, it can also be written as $m=\tfrac{-3}{-4}$ (negative divided by negative is positive!). Go back to $(1,-1)$ and count out the rise, $-3$, and the run, $-4$.

Graph a line using its slope and intercept

We have graphed linear equations by plotting points, using intercepts, recognizing horizontal and vertical lines, and using one point and the slope of the line. Once we see how an equation in slope-intercept form and its graph are related, we’ll have one more method we can use to graph lines.

Let’s look at the graph of $y=\tfrac{1}{2}x+3$ and find its slope and $y$-intercept.

The red lines in the source graph show us the rise is $1$ and the run is $2$. Substituting into the slope formula gives $m=\tfrac{1}{2}$. The $y$-intercept is $(0,3)$.

When a linear equation is solved for $y$, the coefficient of the $x$ term is the slope and the constant term is the $y$-coordinate of the $y$-intercept. We say that $y=\tfrac{1}{2}x+3$ is in slope-intercept form. Sometimes the slope-intercept form is called the “$y$-form.”

Example. Identify the slope and $y$-intercept of the line from each equation: (a) $y=-\tfrac{4}{7}x-2$ (b) $x+3y=9$.

(a) Compare $y=-\tfrac{4}{7}x-2$ to $y=mx+b$. The slope is $m=-\tfrac{4}{7}$ and the $y$-intercept is $(0,-2)$.

(b) When an equation of a line is not given in slope-intercept form, our first step will be to solve the equation for $y$:

The slope is $m=-\tfrac{1}{3}$ and the $y$-intercept is $(0,3)$.

Example. Graph $y=-x+4$ using its slope and $y$-intercept.

The equation is in slope-intercept form. Identify $m=-1$ and the $y$-intercept $(0,4)$. Plot the $y$-intercept. Write $m=\tfrac{-1}{1}$, so the rise is $-1$ and the run is $1$. Count out the rise and run to mark the second point. Draw the line.

Choose the most convenient method to graph a line

Now that we have seen several methods we can use to graph lines, how do we know which method to use for a given equation? While we could plot points, use the slope-intercept form, or find the intercepts for any equation, if we recognize the most convenient way to graph a certain type of equation, our work will be easier. Generally, plotting points is not the most efficient way to graph a line.

Example. Determine the most convenient method to graph each line: (a) $y=5$ (b) $4x-5y=20$ (c) $x=-3$ (d) $y=-\tfrac{5}{9}x+8$.

(a) This equation has only one variable, $y$. Its graph is a horizontal line crossing the $y$-axis at $5$.

(b) This equation is of the form $Ax+By=C$. The easiest way to graph it will be to find the intercepts and one more point.

(c) There is only one variable, $x$. The graph is a vertical line crossing the $x$-axis at $-3$.

(d) Since this equation is in $y=mx+b$ form, it will be easiest to graph this line by using the slope and $y$-intercept.

Graph and interpret applications of slope-intercept

Many real-world applications are modeled by linear equations. We will take a look at a few applications here so you can see how equations written in slope-intercept form relate to real world situations. Usually, when a linear equation models uses real-world data, different letters are used for the variables, instead of using only $x$ and $y$. The variable names remind us of what quantities are being measured. Also, we often will need to extend the axes in our rectangular coordinate system to bigger positive and negative numbers to accommodate the data in the application.

Example. The equation $F=\tfrac{9}{5}C+32$ is used to convert temperatures, $C$, on the Celsius scale to temperatures, $F$, on the Fahrenheit scale.

(a) Find the Fahrenheit temperature for a Celsius temperature of $0$.

(b) Find the Fahrenheit temperature for a Celsius temperature of $20$.

(c) Interpret the slope and $F$-intercept of the equation. Even though this equation uses $F$ and $C$, it is still in slope-intercept form. The slope, $\tfrac{9}{5}$, means that the temperature Fahrenheit ($F$) increases $9$ degrees when the temperature Celsius ($C$) increases $5$ degrees. The $F$-intercept means that when the temperature is $0°$ on the Celsius scale, it is $32°$ on the Fahrenheit scale.

(d) Graph the equation. Start at the $F$-intercept $(0,32)$, and then count out the rise of $9$ and the run of $5$ to get a second point.

The cost of running some types of business has two components—a fixed cost and a variable cost. The fixed cost is always the same regardless of how many units are produced. The variable cost depends on the number of units produced. It is for the material and labor needed to produce each item.

Example. Sam drives a delivery van. The equation $C=0.5m+60$ models the relation between his weekly cost, $C$, in dollars and the number of miles, $m$, that he drives.

(a) Find Sam’s cost for a week when he drives $0$ miles: $C=0.5(0)+60=60$. Sam’s costs are $60 when he drives $0$ miles.

(b) Find the cost for a week when he drives $250$ miles: $C=0.5(250)+60=185$. Sam’s costs are $185 when he drives $250$ miles.

(c) Interpret the slope and $C$-intercept. The slope, $0.5$, means that the weekly cost, $C$, increases by $0.50 when the number of miles driven, $m$, increases by $1$. The $C$-intercept means that when the number of miles driven is $0$, the weekly cost is $60.

(d) Graph the equation. Start at the $C$-intercept $(0,60)$. To count out the slope $m=0.5$, rewrite it as an equivalent fraction: $m=0.5=\tfrac{0.5}{1}=\tfrac{50}{100}$. Go up $50$ from the intercept of $60$ and then right $100$. The second point is $(100,110)$.

Use slopes to identify parallel and perpendicular lines

Two lines that have the same slope are called parallel lines. Parallel lines have the same steepness and never intersect. Two lines that have the same slope and different $y$-intercepts are called parallel lines.

What about vertical lines? The slope of a vertical line is undefined, so vertical lines don’t fit in the definition above. We say that vertical lines that have different $x$-intercepts are parallel.

Example. Use slopes and $y$-intercepts to determine if the lines are parallel: (a) $3x-2y=6$ and $y=\tfrac{3}{2}x+1$ (b) $y=2x-3$ and $-6x+3y=-9$.

(a) Solve the first equation for $y$:

The second line is already $y=\tfrac{3}{2}x+1$. The lines have the same slope and different $y$-intercepts and so they are parallel.

(b) Solving $-6x+3y=-9$ gives $y=2x-3$. The lines have the same slope, but they also have the same $y$-intercepts. Their equations represent the same line and we say the lines are coincident. They are not parallel; they are the same line.

Example. Use slopes and $y$-intercepts to determine if the lines are parallel: (a) $y=-4$ and $y=3$ (b) $x=-2$ and $x=-5$.

(a) These are horizontal lines and so their slopes are both $0$. Their $y$-intercepts are $(0,-4)$ and $(0,3)$. The lines have the same slope and different $y$-intercepts and so they are parallel.

(b) These are vertical lines and their slopes are undefined. They cross the $x$-axis at $x=-2$ and $x=-5$. The lines are vertical and have different $x$-intercepts and so they are parallel.

The lines $y=\tfrac{1}{4}x-1$ and $y=-4x+2$ lie in the same plane and intersect in right angles. We call these lines perpendicular. Their slopes are negative reciprocals of each other, and their product is $-1$:

Example. Use slopes to determine if the lines are perpendicular: (a) $y=-5x-4$ and $x-5y=5$ (b) $7x+2y=3$ and $2x+7y=5$.

(a) The first equation is in slope-intercept form. Solve the second equation for $y$: $x-5y=5$, $-5y=-x+5$, $y=\tfrac{1}{5}x-1$. The slopes are $m_1=-5$ and $m_2=\tfrac{1}{5}$. They are negative reciprocals, so the lines are perpendicular. Since $-5(\tfrac{1}{5})=-1$, it checks.

(b) Solve the equations for $y$: $y=-\tfrac{7}{2}x+\tfrac{3}{2}$ and $y=-\tfrac{2}{7}x+\tfrac{5}{7}$. The slopes are reciprocals of each other, but they have the same sign. Since they are not negative reciprocals, the lines are not perpendicular.

Key terms

slope — the measure of the steepness of a line; the ratio of rise to run. slope formula — $m=\tfrac{y_2-y_1}{x_2-x_1}$, used to find the slope between two points. slope-intercept form — $y=mx+b$, where $m$ is the slope and $(0,b)$ is the $y$-intercept. fixed cost — a business cost that does not change with the number of units produced. variable cost — a business cost that changes with the number of units produced. parallel lines — lines in the same plane that do not intersect. perpendicular lines — lines in the same plane that form a right angle.

This section is adapted from Intermediate Algebra 2e, Section 3.2: 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 coordinate-plane figures as accessible interactive graphs; omitted the Be Prepared quiz, Media links, self-check, and section exercises; and converted the source Try Its into interactive exercises with instant feedback.