86 Slope of a Line
Learning Objectives
By the end of this section, you will be able to:
- Find the slope of a line
- Graph a line given a point and the slope
- Graph a line using its slope and intercept
- Choose the most convenient method to graph a line
- Graph and interpret applications of slope–intercept
- Use slopes to identify parallel and perpendicular lines
Before you get started, take this readiness quiz.
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).
The slope of a line is
The rise measures the vertical change and the run measures the horizontal 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.
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,
- Locate two points on the line whose coordinates are integers.
- Starting with one point, sketch a right triangle, going from the first point to the second point.
- Count the rise and the run on the legs of the triangle.
- Take the ratio of rise to run to find the slope:
Find the slope of the line shown.
| Locate two points on the graph whose
coordinates are integers. |
and |
| Starting at sketch a right triangle to
as shown in this graph. |
|
| Count the rise— since it goes down, it is negative. | The rise is |
| Count the run. | The run is 3. |
| Use the slope formula. | |
| Substitute the values of the rise and run. | |
| Simplify. | |
| The slope of the line is | |
| So y decreases by 2 units as x increases by 3 units. |
Find the slope of the line shown.
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, 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.
Let’s also consider a vertical line, the line as shown in the graph.
The slope is undefined since division by zero is undefined. So we say that the slope of the vertical line 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.
The slope of a horizontal line, is 0.
The slope of a vertical line, is undefined.
Find the slope of each line: ⓐ ⓑ
ⓐ
This is a vertical line. Its slope is undefined.
ⓑ
This is a horizontal line. It has slope 0.
Find the slope of the line:
undefined
Find the slope of the line:
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 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 gives the coordinates of a point. But when we work with slopes, we use two points. How can the same symbol be used to represent two different points? Mathematicians use subscripts to distinguish the points.
We will use to identify the first point and to identify the second point.
If we had more than two points, we could use 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 and as shown in this graph.
We’ve shown that is really another version of We can use this formula to find the slope of a line when we have two points on the line.
The slope of the line between two points and is:
The slope is:
Use the slope formula to find the slope of the line through the points and
Let’s verify this slope on the graph shown.
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Use the slope formula to find the slope of the line through the pair of points: and
Use the slope formula to find the slope of the line through the pair of points: and
10
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.
Graph the line passing through the point whose slope is
You can check your work by finding a third point. Since the slope is it can also be written as (negative divided by negative is positive!). Go back to and count out the rise, and the run,
Graph the line passing through the point with the slope
Graph the line passing through the point with the slope
- Plot the given point.
- Use the slope formula to identify the rise and the run.
- Starting at the given point, count out the rise and run to mark the second point.
- Connect the points with a line.
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). Let’s look at the graph of the equation and find its slope and y-intercept.
The red lines in the graph show us the rise is 1 and the run is 2. Substituting into the slope formula:
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The y-intercept is
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 is in slope–intercept form. Sometimes the slope–intercept form is called the “y-form.”
The slope–intercept form of an equation of a line with slope m and y-intercept, is
Let’s practice finding the values of the slope and y-intercept from the equation of a line.
Identify the slope and y-intercept of the line from the equation:
ⓐⓑ
ⓐ 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. | |
| 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. |
Identify the slope and y-intercept from the equation of the line.
ⓐⓑ
ⓐ
ⓑ
Identify the slope and y-intercept from the equation of the line.
ⓐⓑ
ⓐ
ⓑ
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.
Graph the line of the equation using its slope and y-intercept.
| The equation is in slope–intercept form. | |
| Identify the slope and y-intercept. |
y-intercept is |
| Plot the y-intercept. | See the graph. |
| Identify the rise over the run. | |
| Count out the rise and run to mark the second point. | rise , run 1
|
Draw the line as shown in the graph.
Graph the line of the equation using its slope and y-intercept.
Graph the line of the equation using its slope and y-intercept.
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 We substituted to find the x– intercept and 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.
Consider the form of the equation.
- If it only has one variable, it is a vertical or horizontal line.
- is a vertical line passing through the x-axis at a.
- is a horizontal line passing through the y-axis at b.
- If y is isolated on one side of the equation, in the form graph by using the slope and y-intercept.
- Identify the slope and y-intercept and then graph.
- If the equation is of the form find the intercepts.
- Find the x– and y-intercepts, a third point, and then graph.
Determine the most convenient method to graph each line:
ⓐⓑⓒⓓ
ⓐ
This equation has only one variable, y. Its graph is a horizontal line crossing the y-axis at .
ⓑ
This equation is of the form The easiest way to graph it will be to find the intercepts and one more point.
ⓒ
There is only one variable, x. The graph is a vertical line crossing the x-axis at
ⓓ
Since this equation is in form, it will be easiest to graph this line by using the slope and y-intercepts.
Determine the most convenient method to graph each line:
ⓐⓑⓒⓓ
ⓐ intercepts ⓑ horizontal line ⓒ slope-intercept ⓓ vertical line
Determine the most convenient method to graph each line:
ⓐⓑⓒⓓ
ⓐ vertical line ⓑ slope-intercept ⓒ horizontal line
ⓓ intercepts
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.
The equation 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.
ⓐ
ⓑ
ⓒ
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, 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 on the Celsius scale, it is on the Fahrenheit scale.
ⓓ Graph the equation.
We’ll need to use a larger scale than our usual. Start at the F-intercept , and then count out the rise of 9 and the run of 5 to get a second point as shown in the graph.
The equation is used to estimate a woman’s height in inches, h, based on her shoe size, s.
ⓐ Estimate the height of a child who wears women’s shoe size 0.
ⓑ Estimate the height of a woman with shoe size 8.
ⓒ Interpret the slope and h-intercept of the equation.
ⓓ Graph the equation.
ⓐ 50 inches
ⓑ 66 inches
ⓒ The slope, 2, means that the height, h, increases by 2 inches when the shoe size, s, increases by 1. The h-intercept means that when the shoe size is 0, the height is 50 inches.
ⓓ
The equation is used to estimate the temperature in degrees Fahrenheit, T, based on the number of cricket chirps, n, in one minute.
ⓐ Estimate the temperature when there are no chirps.
ⓑ Estimate the temperature when the number of chirps in one minute is 100.
ⓒ Interpret the slope and T-intercept of the equation.
ⓓ Graph the equation.
ⓐ 40 degrees
ⓑ 65 degrees
ⓒ The slope, means that the temperature Fahrenheit (F) increases 1 degree when the number of chirps, n, increases by 4. The T-intercept means that when the number of chirps is 0, the temperature is 40°.
ⓓ
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.
Sam drives a delivery van. The equation 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.
ⓐ
ⓑ
ⓒ 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 .
To count out the slope m = 0.5, we rewrite it as an equivalent fraction that will make our graphing easier.
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).
Stella has a home business selling gourmet pizzas. The equation models the relation between her weekly cost, C, in dollars and the number of pizzas, p, that she sells.
ⓐ Find Stella’s cost for a week when she sells no pizzas.
ⓑ Find the cost for a week when she sells 15 pizzas.
ⓒ Interpret the slope and C-intercept of the equation.
ⓓ Graph the equation.
ⓐ ?25
ⓑ ?85
ⓒ The slope, 4, means that the weekly cost, C, increases by ?4 when the number of pizzas sold, p, increases by 1. The C-intercept means that when the number of pizzas sold is 0, the weekly cost is ?25.
ⓓ
Loreen has a calligraphy business. The equation models the relation between her weekly cost, C, in dollars and the number of wedding invitations, n, that she writes.
ⓐ Find Loreen’s cost for a week when she writes no invitations.
ⓑ Find the cost for a week when she writes 75 invitations.
ⓒ Interpret the slope and C-intercept of the equation.
ⓓ Graph the equation.
ⓐ ?35
ⓑ ?170
ⓒ The slope, means that the weekly cost, C, increases by when the number of invitations, n, increases by 1.
The C-intercept means that when the number of invitations is 0, the weekly cost is ?35.
ⓓ
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).
Verify that both lines have the same slope, 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.
Parallel lines are lines in the same plane that do not intersect.
- Parallel lines have the same slope and different y-intercepts.
- If and are the slopes of two parallel lines then
- Parallel vertical lines have different x-intercepts.
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.
Use slopes and y-intercepts to determine if the lines are parallel:
ⓐ and ⓑ and
ⓐ
The lines have the same slope and different y-intercepts and so they are parallel.
You may want to graph the lines to confirm whether they are parallel.
ⓑ
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.
Use slopes and y-intercepts to determine if the lines are parallel:
ⓐ and ⓑ and
ⓐ parallel ⓑ not parallel; same line
Use slopes and y-intercepts to determine if the lines are parallel:
ⓐ and ⓑ and
ⓐ parallel ⓑ not parallel; same line
Use slopes and y-intercepts to determine if the lines are parallel:
ⓐ and ⓑ and
ⓐ and
We recognize right away from the equations that these are horizontal lines, and so we know their slopes are both 0.
Since the horizontal lines cross the y-axis at and at we know the y-intercepts are and
The lines have the same slope and different y-intercepts and so they are parallel.
ⓑ and
We recognize right away from the equations that these are vertical lines, and so we know their slopes are undefined.
Since the vertical lines cross the x-axis at and we know the y-intercepts are and
The lines are vertical and have different x-intercepts and so they are parallel.
Use slopes and y-intercepts to determine if the lines are parallel:
ⓐ and ⓑ and
ⓐ parallel ⓑ parallel
Use slopes and y-intercepts to determine if the lines are parallel:
ⓐ and ⓑ and
ⓐ parallel ⓑ parallel
Let’s look at the lines whose equations are and shown in (Figure).
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, and the slope of the second line, we can see that they are negative reciprocals of each other. If we multiply them, their product is
This is always true for perpendicular lines and leads us to this definition.
Perpendicular lines are lines in the same plane that form a right angle.
- If and are the slopes of two perpendicular lines, then:
- their slopes are negative reciprocals of each other,
- the product of their slopes is ,
- A vertical line and a horizontal line are always perpendicular to each other.
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 the lines are perpendicular.
Use slopes to determine if the lines are perpendicular:
ⓐ and ⓑ and
ⓐ
The slopes are negative reciprocals of each other, so the lines are perpendicular. We check by multiplying the slopes, Since it checks.
ⓑ
The slopes are reciprocals of each other, but they have the same sign. Since they are not negative reciprocals, the lines are not perpendicular.
Use slopes to determine if the lines are perpendicular:
ⓐ and ⓑ and
ⓐ perpendicular ⓑ not perpendicular
Use slopes to determine if the lines are perpendicular:
ⓐ and ⓑ and
ⓐ perpendicular ⓑ not perpendicular
Key Concepts
- Slope of a Line
- The slope of a line is
- The rise measures the vertical change and the run measures the horizontal change.
- How to find the slope of a line from its graph using
- Locate two points on the line whose coordinates are integers.
- Starting with one point, sketch a right triangle, going from the first point to the second point.
- Count the rise and the run on the legs of the triangle.
- Take the ratio of rise to run to find the slope:
- Slope of a line between two points.
- The slope of the line between two points and is:
- How to graph a line given a point and the slope.
- Plot the given point.
- Use the slope formula to identify the rise and the run.
- Starting at the given point, count out the rise and run to mark the second point.
- Connect the points with a line.
- Slope Intercept Form of an Equation of a Line
- The slope–intercept form of an equation of a line with slope m and y-intercept, is
- Parallel Lines
- Parallel lines are lines in the same plane that do not intersect.
Parallel lines have the same slope and different y-intercepts.
If and are the slopes of two parallel lines then
Parallel vertical lines have different x-intercepts.
- Parallel lines are lines in the same plane that do not intersect.
- Perpendicular Lines
- Perpendicular lines are lines in the same plane that form a right angle.
- If and are the slopes of two perpendicular lines, then:
their slopes are negative reciprocals of each other,
the product of their slopes is
- A vertical line and a horizontal line are always perpendicular to each other.
Practice Makes Perfect
Find the Slope of a Line
In the following exercises, find the slope of each line shown.
In the following exercises, find the slope of each line.
0
undefined
In the following exercises, use the slope formula to find the slope of the line between each pair of points.
Graph a Line Given a Point and the Slope
In the following exercises, graph each line with the given point and slope.
;
;
;
y-intercept 3;
x-intercept ;
;
;
Graph a Line Using Its Slope and Intercept
In the following exercises, identify the slope and y-intercept of each line.
In the following exercises, graph the line of each equation using its slope and y-intercept.
Choose the Most Convenient Method to Graph a Line
In the following exercises, determine the most convenient method to graph each line.
vertical line
slope-intercept
intercepts
intercepts
Graph and Interpret Applications of Slope–Intercept
The equation models the relation between the amount of Tuyet’s monthly water bill payment, P, in dollars, and the number of units of water, w, used.
ⓐ Find Tuyet’s payment for a month when 0 units of water are used.
ⓑ Find Tuyet’s payment for a month when 12 units of water are used.
ⓒ Interpret the slope and P-intercept of the equation.
ⓓ Graph the equation.
ⓐ ?31
ⓑ ?52
ⓒ The slope, means that the payment, P, increases by when the number of units of water used, w, increases by 1. The P-intercept means that when the number units of water Tuyet used is 0, the payment is ?31.
ⓓ
The equation models the relation between the amount of R and y’s monthly water bill payment, P, in dollars, and the number of units of water, w, used.
ⓐ Find the payment for a month when R and y used 0 units of water.
ⓑ Find the payment for a month when R and y used 15 units of water.
ⓒ Interpret the slope and P-intercept of the equation.
ⓓ Graph the equation.
Bruce drives his car for his job. The equation models the relation between the amount in dollars, R, that he is reimbursed and the number of miles, m, he drives in one day.
ⓐ Find the amount Bruce is reimbursed on a day when he drives 0 miles.
ⓑ Find the amount Bruce is reimbursed on a day when he drives 220 miles.
ⓒ Interpret the slope and R-intercept of the equation.
ⓓ Graph the equation.
ⓐ ?42
ⓑ ?168.50
ⓒ The slope, 0.575 means that the amount he is reimbursed, R, increases by ?0.575 when the number of miles driven, m, increases by 1. The R-intercept means that when the number miles driven is 0, the amount reimbursed is ?42.
ⓓ
Janelle is planning to rent a car while on vacation. The equation models the relation between the cost in dollars, C, per day and the number of miles, m, she drives in one day.
ⓐ Find the cost if Janelle drives the car 0 miles one day.
ⓑ Find the cost on a day when Janelle drives the car 400 miles.
ⓒ Interpret the slope and C-intercept of the equation.
ⓓ Graph the equation.
Cherie works in retail and her weekly salary includes commission for the amount she sells. The equation models the relation between her weekly salary, S, in dollars and the amount of her sales, c, in dollars.
ⓐ Find Cherie’s salary for a week when her sales were ?0.
ⓑ Find Cherie’s salary for a week when her sales were ?3,600.
ⓒ Interpret the slope and S-intercept of the equation.
ⓓ Graph the equation.
ⓐ ?400
ⓑ ?940
ⓒ The slope, means that Cherie’s salary, S, increases by ?0.15 for every ?1 increase in her sales. The S-intercept means that when her sales are ?0, her salary is ?400.
ⓓ
Patel’s weekly salary includes a base pay plus commission on his sales. The equation models the relation between his weekly salary, S, in dollars and the amount of his sales, c, in dollars.
ⓐ Find Patel’s salary for a week when his sales were 0.
ⓑ Find Patel’s salary for a week when his sales were 18,540.
ⓒ Interpret the slope and S-intercept of the equation.
ⓓ Graph the equation.
Costa is planning a lunch banquet. The equation models the relation between the cost in dollars, C, of the banquet and the number of guests, g.
ⓐ Find the cost if the number of guests is 40.
ⓑ Find the cost if the number of guests is 80.
ⓒ Interpret the slope and C-intercept of the equation.
ⓓ Graph the equation.
ⓐ ?1570
ⓑ ?5690
ⓒ The slope gives the cost per guest. The slope, 28, means that the cost, C, increases by ?28 when the number of guests increases by 1. The C-intercept means that if the number of guests was 0, the cost would be ?450.
ⓓ
Margie is planning a dinner banquet. The equation models the relation between the cost in dollars, C of the banquet and the number of guests, g.
ⓐ Find the cost if the number of guests is 50.
ⓑ Find the cost if the number of guests is 100.
ⓒ Interpret the slope and C-intercept of the equation.
ⓓ Graph the equation.
Use Slopes to Identify Parallel and Perpendicular Lines
In the following exercises, use slopes and y-intercepts to determine if the lines are parallel, perpendicular, or neither.
parallel
neither
parallel
perpendicular
neither
perpendicular
perpendicular
neither
parallel
Writing Exercises
How does the graph of a line with slope differ from the graph of a line with slope
Answers will vary.
Why is the slope of a vertical line “undefined”?
Explain how you can graph a line given a point and its slope.
Answers will vary.
Explain in your own words how to decide which method to use to graph a line.
Self Check
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ After reviewing this checklist, what will you do to become confident for all objectives?
Glossary
- parallel lines
- Parallel lines are lines in the same plane that do not intersect.
- perpendicular lines
- Perpendicular lines are lines in the same plane that form a right angle.