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, as shown in this illustration. Slope is a rate of change. See Figure 3.5.

Figure 3.5

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.

To find the slope of the line, 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 3.12

Find the slope of the line shown.

Show solution

Solution

Locate two points on the graph whose coordinates are integers.	$\left(0,5\right)$ and $\left(3,3\right)$
Starting at $\left(0,5\right),$ sketch a right triangle to $\left(3,3\right)$ as shown in this graph.
Count the rise— since it goes down, it is negative.	The rise is $\mathrm{-2}.$
Count the run.	The run is 3.
Use the slope formula.	$m=\frac{\text{rise}}{\text{run}}$
Substitute the values of the rise and run.	$m=\frac{\mathrm{-2}}{3}$
Simplify.	$m=-\frac{2}{3}$
	The slope of the line is $-\frac{2}{3}.$
	So y decreases by 2 units as x increases by 3 units.

Try It 3.23

Find the slope of the line shown.

Try It 3.24

Find the slope of the line shown.

How do we find the slope of horizontal and vertical lines? To find the slope of the horizontal line, $y=4,$ we could graph the line, find two points on it, and count the rise and the run. Let’s see what happens when we do this, as shown in the graph below.

$\begin{array}{cc}\text{What is the rise?}\hfill & \text{The rise is 0.}\hfill \\ \text{What is the run?}\hfill & \text{The run is 3.}\hfill \\ \text{What is the slope?}\hfill & m=\frac{\text{rise}}{\text{run}}\hfill \\ & m=\frac{0}{3}\hfill \\ & m=0\hfill \\ & \text{The slope of the horizontal line}y=4\text{is 0.}\hfill \end{array}$

Let’s also consider a vertical line, the line $x=3,$ as shown in the graph.

$\begin{array}{cc}\text{What is the rise?}\hfill & \text{The rise is 2.}\hfill \\ \\ \text{What is the run?}\hfill & \text{The run is 0.}\hfill \\ \\ \text{What is the slope?}\hfill & m=\frac{\text{rise}}{\text{run}}\hfill \\ & m=\frac{2}{0}\hfill \end{array}$

The slope is undefined since division by zero is undefined. So we say that the slope of the vertical line $x=3$ is undefined.

All horizontal lines have slope 0. When the y-coordinates are the same, the rise is 0.

The slope of any vertical line is undefined. When the x-coordinates of a line are all the same, the run is 0.

Quick Guide to the Slopes of Lines

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 as we’ll see, there is a way to find the slope without graphing. Before we get to it, we need to introduce some algebraic notation.

We have seen that an ordered pair $\left(x,y\right)$ gives the coordinates of a point. But when we work with slopes, we use two points. How can the same symbol $\left(x,y\right)$ be used to represent two different points? Mathematicians use subscripts to distinguish the points.

We will use $\left(x_1,y_1\right)$ to identify the first point and $\left(x_2,y_2\right)$ to identify the second point.

If we had more than two points, we could use $\left(x_3,y_3\right),$$\left(x_4,y_4\right),$ and so on.

Let’s see how the rise and run relate to the coordinates of the two points by taking another look at the slope of the line between the points $\left(2,3\right)$ and $\left(7,6\right),$ as shown in this graph.

$\begin{array}{cccc}\text{Since we have two points, we will use subscript notation.}\hfill & & & \left(\stackrel{x_1,}{2,}\stackrel{y_1}{3}\right)\left(\stackrel{x_2,y_2}{7,6}\right)\hfill \\ & & & m=\frac{\text{rise}}{\text{run}}\hfill \\ \text{On the graph, we counted the rise of 3 and the run of 5.}\hfill & & & m=\frac{3}{5}\hfill \\ \text{Notice that the rise of 3 can be found by subtracting the}\hfill & & & \\ y\text{-coordinates, 6 and 3, and the run of 5 can be found by}\hfill & & & \\ \text{subtracting the}x\text{-coordinates 7 and 2.}\hfill & & & \\ \text{We rewrite the rise and run by putting in the coordinates.}\hfill & & & m=\frac{6-3}{7-2}\hfill \\ \\ \\ \text{But 6 is}{y}_2,\text{the}y\text{-coordinate of the second point and 3 is}{y}_1,\text{the}y\text{-coordinate}\hfill & & & \\ \text{of the first point. So we can rewrite the slope using subscript notation.}\hfill & & & m=\frac{y_2-y_1}{7-2}\hfill \\ \\ \\ \text{Also 7 is the}x\text{-coordinate of the second point and 2 is the}x\text{-coordinate}\hfill & & & \\ \text{of the first point. So again we rewrite the slope using subscript notation.}\hfill & & & m=\frac{y_2-y_1}{x_2-x_1}\hfill \\ \\ \end{array}$

We’ve shown that $m=\frac{y_2-y_1}{x_2-x_1}$ is really another version of $m=\frac{\text{rise}}{\text{run}}.$ We can use this formula to find the slope of a line when we have two points on the line.

The slope is:

Example 3.14

Use the slope formula to find the slope of the line through the points $\left(\mathrm{-2},\mathrm{-3}\right)$ and $\left(\mathrm{-7},4\right).$

Show solution

Solution

$\begin{array}{cc}\text{We’ll call}\left(\mathrm{-2},\mathrm{-3}\right)\text{point}\text{\#}1\text{and}\left(\mathrm{-7},4\right)\text{point}\text{\#}2.\hfill & \left(\begin{array}{cc}{x}_1,\hfill & y_1\hfill \\ -2,\hfill & -3\hfill \end{array}\right)\left(\begin{array}{cc}{x}_2,\hfill & y_2\hfill \\ -7,\hfill & 4\hfill \end{array}\right)\hfill \\ \text{Use the slope formula.}\hfill & m=\frac{y_2-y_1}{x_2-x_1}\hfill \\ \text{Substitute the values.}\hfill & \\ y\text{of the second point minus}y\text{of the first point}\hfill \\ x\text{of the second point minus}x\text{of the first point}\hfill & m=\frac{4-(\mathrm{-3})}{\mathrm{-7}-(\mathrm{-2})}\hfill \\ \text{Simplify.}\hfill & \begin{array}{c}m=\frac{7}{\mathrm{-5}}\hfill \\ m=-\frac{7}{5}\hfill \end{array}\hfill \end{array}$
Let’s verify this slope on the graph shown.

$$\begin{array}{}\\ \\ m=\frac{\text{rise}}{\text{run}}\hfill \\ m=\frac{7}{\mathrm{-5}}\hfill \\ m=-\frac{7}{5}\hfill \end{array}$$

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 3.15

How to graph a Line Given a Point and the Slope

Graph the line passing through the point $\left(1,\mathrm{-1}\right)$ whose slope is $m=\frac{3}{4}.$

Show solution

Solution

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

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.

See Figure 3.6. Let’s look at the graph of the equation $y=\frac{1}{2}x+3$ and find its slope and y-intercept.

Figure 3.6

The red lines in the graph show us the rise is 1 and the run is 2. Substituting into the slope formula:

The y-intercept is $\left(0,3\right).$

Look at the equation of this line.

Look at the slope and y-intercept.

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 the equation $y=\frac{1}{2}x+3$ is in slope–intercept form. Sometimes the slope–intercept form is called the “y-form.”

Let’s practice finding the values of the slope and y-intercept from the equation of a line.

Example 3.16

Identify the slope and y-intercept of the line from the equation:

ⓐ $y=-\frac{4}{7}x-2$ ⓑ $x+3y=9$

Show solution

Solution

ⓐ We compare our equation to the slope–intercept form of the equation.

Write the slope–intercept form of the equation of the line.
Write the equation of the line.
Identify the slope.
Identify the y-intercept.

ⓑ When an equation of a line is not given in slope–intercept form, our first step will be to solve the equation for y.

Solve for y.	$x+3y=9$
Subtract x from each side.
Divide both sides by 3.
Simplify.
Write the slope–intercept form of the equation of the line.
Write the equation of the line.
Identify the slope.
Identify the y-intercept.

We have graphed a line using the slope and a point. Now that we know how to find the slope and y-intercept of a line from its equation, we can use the y-intercept as the point, and then count out the slope from there.

Example 3.17

Graph the line of the equation $y=\text{−}x+4$ using its slope and y-intercept.

Show solution

Solution

	$y=mx+b$
The equation is in slope–intercept form.	$y=\text{−}x+4$
Identify the slope and y-intercept.	$m=\mathrm{-1}$ y-intercept is $\left(0,4\right)$
Plot the y-intercept.	See the graph.
Identify the rise over the run.	$m=\frac{\mathrm{-1}}{1}$
Count out the rise and run to mark the second point.	rise $\mathrm{-1}$, run 1

Draw the line as shown in the graph.

Now that we have graphed lines by using the slope and y-intercept, let’s summarize all the methods we have used to graph lines.

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. Let’s look for some patterns to help determine the most convenient method to graph a line.

Here are five equations we graphed in this chapter, and the method we used to graph each of them.

Equations #1 and #2 each have just one variable. Remember, in equations of this form the value of that one variable is constant; it does not depend on the value of the other variable. Equations of this form have graphs that are vertical or horizontal lines.

In equations #3 and #4, both x and y are on the same side of the equation. These two equations are of the form $Ax+By=C.$ We substituted $y=0$ to find the x- intercept and $x=0$ to find the y-intercept, and then found a third point by choosing another value for x or y.

Equation #5 is written in slope–intercept form. After identifying the slope and y-intercept from the equation we used them to graph the line.

This leads to the following strategy.

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 3.19

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

ⓐ Find the Fahrenheit temperature for a Celsius temperature of 0.

ⓑ Find the Fahrenheit temperature for a Celsius temperature of 20.

ⓒ Interpret the slope and F-intercept of the equation.

ⓓ Graph the equation.

Show solution

Solution

ⓐ
$\begin{array}{cc}\text{Find the Fahrenheit temperature for a Celsius temperature of 0.}\hfill & F=\frac{9}{5}C+32\hfill \\ \text{Find}F\text{when}C=0.\hfill & F=\frac{9}{5}\left(0\right)+32\hfill \\ \text{Simplify.}\hfill & F=32\hfill \end{array}$
ⓑ
$\begin{array}{cc}\text{Find the Fahrenheit temperature for a Celsius temperature of 20.}\hfill & F=\frac{9}{5}C+32\hfill \\ \text{Find}F\text{when}C=20.\hfill & F=\frac{9}{5}\left(20\right)+32\hfill \\ \text{Simplify.}\hfill & F=36+32\hfill \\ \text{Simplify.}\hfill & F=68\hfill \end{array}$
ⓒ
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, $\frac{9}{5},$ 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 $0\text{°}$ on the Celsius scale, it is $32\text{°}$ on the Fahrenheit scale.
ⓓ Graph the equation.
We’ll need to use a larger scale than our usual. Start at the F-intercept $\left(0,32\right)$, and then count out the rise of 9 and the run of 5 to get a second point as shown in the graph.

The cost of running some types business have two components—a fixed cost and a variable cost. The fixed cost is always the same regardless of how many units are produced. This is the cost of rent, insurance, equipment, advertising, and other items that must be paid regularly. The variable cost depends on the number of units produced. It is for the material and labor needed to produce each item.

Example 3.20

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.

ⓐ Find Sam’s cost for a week when he drives 0 miles.

ⓑ Find the cost for a week when he drives 250 miles.

ⓒ Interpret the slope and C-intercept of the equation.

ⓓ Graph the equation.

Show solution

Solution

ⓐ
$\begin{array}{cccc}\text{Find Sam’s cost for a week when he drives 0 miles.}\hfill & & & C=0.5m+60\hfill \\ \text{Find}C\text{when}m=0.\hfill & & & C=0.5\left(0\right)+60\hfill \\ \text{Simplify.}\hfill & & & C=60\hfill \\ & & & \text{Sam’s costs are \$60 when he drives 0 miles.}\hfill \end{array}$
ⓑ
$\begin{array}{cccc}\text{Find the cost for a week when he drives 250 miles.}\hfill & & & C=0.5m+60\hfill \\ \text{Find}C\text{when}m=250.\hfill & & & C=0.5\left(250\right)+60\hfill \\ \text{Simplify.}\hfill & & & C=185\hfill \\ & & & \text{Sam’s costs are \$185 when he drives 250 miles.}\hfill \end{array}$
ⓒ Interpret the slope and C-intercept of the equation.

The slope, 0.5, means that the weekly cost, C, increases by $0.50 when the number of miles driven, n, increases by 1.
The C-intercept means that when the number of miles driven is 0, the weekly cost is $60.
ⓓ Graph the equation.
We’ll need to use a larger scale than our usual. Start at the C-intercept $\left(0,60\right)$.

To count out the slope m = 0.5, we rewrite it as an equivalent fraction that will make our graphing easier.

$\begin{array}{cccc}& & & m=0.5\hfill \\ \text{Rewrite as a fraction.}\hfill & & & m=\frac{0.5}{1}\hfill \\ \text{Multiply numerator and}\hfill & & & \\ \text{denominator by}100.\hfill & & & m=\frac{0.5\left(100\right)}{1\left(100\right)}\hfill \\ \text{Simplify.}\hfill & & & m=\frac{50}{100}\hfill \end{array}$

So to graph the next point go up 50 from the intercept of 60 and then to the right 100. The second point will be (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.

We say this more formally in terms of the rectangular coordinate system. Two lines that have the same slope and different y-intercepts are called parallel lines. See Figure 3.7.

Figure 3.7

Verify that both lines have the same slope, $m=\frac{2}{5},$ and different y-intercepts.

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, like the lines shown in this graph.

Figure 3.8

Since parallel lines have the same slope and different y-intercepts, we can now just look at the slope–intercept form of the equations of lines and decide if the lines are parallel.

Let’s look at the lines whose equations are $y=\frac{1}{4}x-1$ and $y=\mathrm{-4}x+2,$ shown in Figure 3.9.

Figure 3.9

These lines lie in the same plane and intersect in right angles. We call these lines perpendicular.

If we look at the slope of the first line, $m_1=\frac{1}{4},$ and the slope of the second line, $m_2=\mathrm{-4},$ we can see that they are negative reciprocals of each other. If we multiply them, their product is $\mathrm{-1}.$

This is always true for perpendicular lines and leads us to this definition.

We were able to look at the slope–intercept form of linear equations and determine whether or not the lines were parallel. We can do the same thing for perpendicular lines.

We find the slope–intercept form of the equation, and then see if the slopes are opposite reciprocals. If the product of the slopes is $\mathrm{-1},$ the lines are perpendicular.