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Elementary Algebra 2e

4.5 Use the Slope-Intercept Form of an Equation of a Line

Elementary Algebra 2e4.5 Use the Slope-Intercept Form of an Equation of a Line
  1. Preface
  2. 1 Foundations
    1. Introduction
    2. 1.1 Introduction to Whole Numbers
    3. 1.2 Use the Language of Algebra
    4. 1.3 Add and Subtract Integers
    5. 1.4 Multiply and Divide Integers
    6. 1.5 Visualize Fractions
    7. 1.6 Add and Subtract Fractions
    8. 1.7 Decimals
    9. 1.8 The Real Numbers
    10. 1.9 Properties of Real Numbers
    11. 1.10 Systems of Measurement
    12. Key Terms
    13. Key Concepts
    14. Exercises
      1. Review Exercises
      2. Practice Test
  3. 2 Solving Linear Equations and Inequalities
    1. Introduction
    2. 2.1 Solve Equations Using the Subtraction and Addition Properties of Equality
    3. 2.2 Solve Equations using the Division and Multiplication Properties of Equality
    4. 2.3 Solve Equations with Variables and Constants on Both Sides
    5. 2.4 Use a General Strategy to Solve Linear Equations
    6. 2.5 Solve Equations with Fractions or Decimals
    7. 2.6 Solve a Formula for a Specific Variable
    8. 2.7 Solve Linear Inequalities
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  4. 3 Math Models
    1. Introduction
    2. 3.1 Use a Problem-Solving Strategy
    3. 3.2 Solve Percent Applications
    4. 3.3 Solve Mixture Applications
    5. 3.4 Solve Geometry Applications: Triangles, Rectangles, and the Pythagorean Theorem
    6. 3.5 Solve Uniform Motion Applications
    7. 3.6 Solve Applications with Linear Inequalities
    8. Key Terms
    9. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  5. 4 Graphs
    1. Introduction
    2. 4.1 Use the Rectangular Coordinate System
    3. 4.2 Graph Linear Equations in Two Variables
    4. 4.3 Graph with Intercepts
    5. 4.4 Understand Slope of a Line
    6. 4.5 Use the Slope-Intercept Form of an Equation of a Line
    7. 4.6 Find the Equation of a Line
    8. 4.7 Graphs of Linear Inequalities
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  6. 5 Systems of Linear Equations
    1. Introduction
    2. 5.1 Solve Systems of Equations by Graphing
    3. 5.2 Solving Systems of Equations by Substitution
    4. 5.3 Solve Systems of Equations by Elimination
    5. 5.4 Solve Applications with Systems of Equations
    6. 5.5 Solve Mixture Applications with Systems of Equations
    7. 5.6 Graphing Systems of Linear Inequalities
    8. Key Terms
    9. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  7. 6 Polynomials
    1. Introduction
    2. 6.1 Add and Subtract Polynomials
    3. 6.2 Use Multiplication Properties of Exponents
    4. 6.3 Multiply Polynomials
    5. 6.4 Special Products
    6. 6.5 Divide Monomials
    7. 6.6 Divide Polynomials
    8. 6.7 Integer Exponents and Scientific Notation
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  8. 7 Factoring
    1. Introduction
    2. 7.1 Greatest Common Factor and Factor by Grouping
    3. 7.2 Factor Trinomials of the Form x2+bx+c
    4. 7.3 Factor Trinomials of the Form ax2+bx+c
    5. 7.4 Factor Special Products
    6. 7.5 General Strategy for Factoring Polynomials
    7. 7.6 Quadratic Equations
    8. Key Terms
    9. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  9. 8 Rational Expressions and Equations
    1. Introduction
    2. 8.1 Simplify Rational Expressions
    3. 8.2 Multiply and Divide Rational Expressions
    4. 8.3 Add and Subtract Rational Expressions with a Common Denominator
    5. 8.4 Add and Subtract Rational Expressions with Unlike Denominators
    6. 8.5 Simplify Complex Rational Expressions
    7. 8.6 Solve Rational Equations
    8. 8.7 Solve Proportion and Similar Figure Applications
    9. 8.8 Solve Uniform Motion and Work Applications
    10. 8.9 Use Direct and Inverse Variation
    11. Key Terms
    12. Key Concepts
    13. Exercises
      1. Review Exercises
      2. Practice Test
  10. 9 Roots and Radicals
    1. Introduction
    2. 9.1 Simplify and Use Square Roots
    3. 9.2 Simplify Square Roots
    4. 9.3 Add and Subtract Square Roots
    5. 9.4 Multiply Square Roots
    6. 9.5 Divide Square Roots
    7. 9.6 Solve Equations with Square Roots
    8. 9.7 Higher Roots
    9. 9.8 Rational Exponents
    10. Key Terms
    11. Key Concepts
    12. Exercises
      1. Review Exercises
      2. Practice Test
  11. 10 Quadratic Equations
    1. Introduction
    2. 10.1 Solve Quadratic Equations Using the Square Root Property
    3. 10.2 Solve Quadratic Equations by Completing the Square
    4. 10.3 Solve Quadratic Equations Using the Quadratic Formula
    5. 10.4 Solve Applications Modeled by Quadratic Equations
    6. 10.5 Graphing Quadratic Equations in Two Variables
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  12. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
    8. Chapter 8
    9. Chapter 9
    10. Chapter 10
  13. Index
Be Prepared 4.10

Before you get started, take this readiness quiz.

Add: x4+14.x4+14.
If you missed this problem, review Example 1.77.

Be Prepared 4.11

Find the reciprocal of 37.37.
If you missed this problem, review Example 1.70.

Be Prepared 4.12

Solve 2x3y=12fory2x3y=12fory.
If you missed this problem, review Example 2.63.

Recognize the Relation Between the Graph and the Slope–Intercept Form of an Equation of a Line

We have graphed linear equations by plotting points, using intercepts, recognizing horizontal and vertical lines, and using the point–slope method. 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.

In Graph Linear Equations in Two Variables, we graphed the line of the equation y=12x+3y=12x+3 by plotting points. See Figure 4.24. Let’s find the slope of this line.

This figure shows a line graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 8 to 8. The y-axis of the plane runs from negative 8 to 8. The line is labeled with the equation y equals one half x, plus 3. The points (0, 3), (2, 4) and (4, 5) are labeled also. A red vertical line begins at the point (2, 4) and ends one unit above the point. It is labeled “Rise equals 1”. A red horizontal line begins at the end of the vertical line and ends at the point (4, 5). It is labeled “Run equals 2. The red lines create a right triangle with the line y equals one half x, plus 3 as the hypotenuse.
Figure 4.24

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

m=riserunm=12m=riserunm=12

What is the y-intercept of the line? The y-intercept is where the line crosses the y-axis, so y-intercept is (0,3)(0,3). The equation of this line is:

The figure shows the equation y equals one half x, plus 3. The fraction one half is colored red and the number 3 is colored blue.

Notice, the line has:

The figure shows the statement “slope m equals one half and y-intercept (0, 3). The slope, one half, is colored red and the number 3 in the y-intercept is colored blue.

When a linear equation is solved for yy, the coefficient of the xx term is the slope and the constant term is the y-coordinate of the y-intercept. We say that the equation y=12x+3y=12x+3 is in slope–intercept form.

The figure shows the statement “m equals one half; y-intercept is (0, 3). The slope, one half, is colored red and the number 3 in the y-intercept is colored blue. Below that statement is the equation y equals one half x, plus 3. The fraction one half is colored red and the number 3 is colored blue. Below the equation is another equation y equals m x, plus b. The variable m is colored red and the variable b is colored blue.

Slope-Intercept Form of an Equation of a Line

The slope–intercept form of an equation of a line with slope mm and y-intercept, (0,b)(0,b) is,

y=mx+by=mx+b

Sometimes the slope–intercept form is called the “y-form.”

Example 4.40

Use the graph to find the slope and y-intercept of the line, y=2x+1y=2x+1.

Compare these values to the equationy=mx+by=mx+b.

Try It 4.79

Use the graph to find the slope and y-intercept of the line y=23x1y=23x1. Compare these values to the equation y=mx+by=mx+b.

The figure shows a line graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 8 to 8. The y-axis of the plane runs from negative 8 to 8. The line goes through the points (0, negative 1) and (6, 3).
Try It 4.80

Use the graph to find the slope and y-intercept of the line y=12x+3y=12x+3. Compare these values to the equation y=mx+by=mx+b.

The figure shows a line graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 8 to 8. The y-axis of the plane runs from negative 8 to 8. The line goes through the points (0, 3) and (negative 6, 0).

Identify the Slope and y-Intercept From an Equation of a Line

In Understand Slope of a Line, we graphed a line using the slope and a point. When we are given an equation in slope–intercept form, we can use the y-intercept as the point, and then count out the slope from there. Let’s practice finding the values of the slope and y-intercept from the equation of a line.

Example 4.41

Identify the slope and y-intercept of the line with equation y=−3x+5y=−3x+5.

Try It 4.81

Identify the slope and y-intercept of the line y=25x1y=25x1.

Try It 4.82

Identify the slope and y-intercept of the line y=43x+1y=43x+1.

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

Example 4.42

Identify the slope and y-intercept of the line with equation x+2y=6x+2y=6.

Try It 4.83

Identify the slope and y-intercept of the line x+4y=8x+4y=8.

Try It 4.84

Identify the slope and y-intercept of the line 3x+2y=123x+2y=12.

Graph a Line Using its Slope and Intercept

Now that we know how to find the slope and y-intercept of a line from its equation, we can graph the line by plotting the y-intercept and then using the slope to find another point.

Example 4.43 How to Graph a Line Using its Slope and Intercept

Graph the line of the equation y=4x2y=4x2 using its slope and y-intercept.

Try It 4.85

Graph the line of the equation y=4x+1y=4x+1 using its slope and y-intercept.

Try It 4.86

Graph the line of the equation y=2x3y=2x3 using its slope and y-intercept.

How To

Graph a line using its slope and y-intercept.

  1. Step 1. Find the slope-intercept form of the equation of the line.
  2. Step 2. Identify the slope and y-intercept.
  3. Step 3. Plot the y-intercept.
  4. Step 4. Use the slope formula m=riserunm=riserun to identify the rise and the run.
  5. Step 5. Starting at the y-intercept, count out the rise and run to mark the second point.
  6. Step 6. Connect the points with a line.

Example 4.44

Graph the line of the equation y=x+4y=x+4 using its slope and y-intercept.

Try It 4.87

Graph the line of the equation y=x3y=x3 using its slope and y-intercept.

Try It 4.88

Graph the line of the equation y=x1y=x1 using its slope and y-intercept.

Example 4.45

Graph the line of the equation y=23x3y=23x3 using its slope and y-intercept.

Try It 4.89

Graph the line of the equation y=52x+1y=52x+1 using its slope and y-intercept.

Try It 4.90

Graph the line of the equation y=34x2y=34x2 using its slope and y-intercept.

Example 4.46

Graph the line of the equation 4x3y=124x3y=12 using its slope and y-intercept.

Try It 4.91

Graph the line of the equation 2xy=62xy=6 using its slope and y-intercept.

Try It 4.92

Graph the line of the equation 3x2y=83x2y=8 using its slope and y-intercept.

We have used a grid with xx and yy both going from about −10−10 to 10 for all the equations we’ve graphed so far. Not all linear equations can be graphed on this small grid. Often, especially in applications with real-world data, we’ll need to extend the axes to bigger positive or smaller negative numbers.

Example 4.47

Graph the line of the equation y=0.2x+45y=0.2x+45 using its slope and y-intercept.

Try It 4.93

Graph the line of the equation y=0.5x+25y=0.5x+25 using its slope and y-intercept.

Try It 4.94

Graph the line of the equation y=0.1x30y=0.1x30 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. See Figure 4.25.

The table has two rows and four columns. The first row spans all four columns and is a header row. The header is “Methods to Graph Lines”. The second row is made up of four columns. The first column is labeled “Plotting Points” and shows a smaller table with four rows and two columns. The first row is a header row with the first column labeled “x” and the second labeled “y”. The rest of the table is blank. Below the table it reads “Find three points. Plot the points, make sure they line up, then draw the line.” The Second column is labeled “Slope–Intercept” and shows the equation y equals m x, plus b. Below the equation it reads “Find the slope and y-intercept. Start at the y-intercept, then count the slope to get a second point.” The third column is labeled “Intercepts” and shows a smaller table with four rows and two columns. The first row is a header row with the first column labeled “x” and the second labeled “y”. The second row has a 0 in the “x” column and the “y” column is blank. The second row is blank in the “x” column and has a 0 in the “y” column. The third row is blank. Below the table it reads “Find the intercepts and a third point. Plot the points, make sure they line up, then draw the line.” The fourth column is labeled “Recognize Vertical and Horizontal Lines”. Below that it reads “The equation has only one variable.” The equation x equals a is a vertical line and the equation y equals b is a horizontal line.
Figure 4.25

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. We saw better methods in sections 4.3, 4.4, and earlier in this section. Let’s look for some patterns to help determine the most convenient method to graph a line.

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

EquationMethod#1x=2Vertical line#2y=4Horizontal line#3x+2y=6Intercepts#44x3y=12Intercepts#5y=4x2Slope–intercept#6y=x+4Slope–interceptEquationMethod#1x=2Vertical line#2y=4Horizontal line#3x+2y=6Intercepts#44x3y=12Intercepts#5y=4x2Slope–intercept#6y=x+4Slope–intercept

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

Equations #5 and #6 are 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.

Strategy for Choosing the Most Convenient Method to Graph a Line

Consider the form of the equation.

  • If it only has one variable, it is a vertical or horizontal line.
    • x=ax=a is a vertical line passing through the x-axis at aa.
    • y=by=b is a horizontal line passing through the y-axis at bb.
  • If yy is isolated on one side of the equation, in the form y=mx+by=mx+b, graph by using the slope and y-intercept.
    • Identify the slope and y-intercept and then graph.
  • If the equation is of the form Ax+By=CAx+By=C, find the intercepts.
    • Find the x- and y-intercepts, a third point, and then graph.

Example 4.48

Determine the most convenient method to graph each line.

y=−6y=−6 5x3y=155x3y=15 x=7x=7 y=25x1y=25x1.

Try It 4.95

Determine the most convenient method to graph each line: 3x+2y=123x+2y=12 y=4y=4 y=15x4y=15x4 x=−7x=−7.

Try It 4.96

Determine the most convenient method to graph each line: x=6x=6 y=34x+1y=34x+1 y=−8y=−8 4x3y=−14x3y=−1.

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 a real-world situation, different letters are used for the variables, instead of x and y. The variable names remind us of what quantities are being measured.

Example 4.49

The equation F=95C+32F=95C+32 is used to convert temperatures, CC, on the Celsius scale to temperatures, FF, 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.

Try It 4.97

The equation h=2s+50h=2s+50 is used to estimate a woman’s height in inches, h, based on her shoe size, s.

  1. Estimate the height of a child who wears women’s shoe size 0.
  2. Estimate the height of a woman with shoe size 8.
  3. Interpret the slope and h-intercept of the equation.
  4. Graph the equation.
Try It 4.98

The equation T=14n+40T=14n+40 is used to estimate the temperature in degrees Fahrenheit, T, based on the number of cricket chirps, n, in one minute.

  1. Estimate the temperature when there are no chirps.
  2. Estimate the temperature when the number of chirps in one minute is 100.
  3. Interpret the slope and T-intercept of the equation.
  4. Graph the equation.

The cost of running some types 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. 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 4.50

Stella has a home business selling gourmet pizzas. The equation C=4p+25C=4p+25 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.

Try It 4.99

Sam drives a delivery van. The equation C=0.5m+60C=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.

Try It 4.100

Loreen has a calligraphy business. The equation C=1.8n+35C=1.8n+35 models the relation between her weekly cost, C, in dollars and the number of wedding invitations, n, that she writes.

  1. Find Loreen’s cost for a week when she writes no invitations.
  2. Find the cost for a week when she writes 75 invitations.
  3. Interpret the slope and C-intercept of the equation.
  4. Graph the equation.

Use Slopes to Identify Parallel Lines

The slope of a line indicates how steep the line is and whether it rises or falls as we read it from left to right. Two lines that have the same slope are called parallel lines. Parallel lines never intersect.

The figure shows three pairs of lines side-by-side. The pair of lines on the left run diagonally rising from left to right. The pair run side-by-side, not crossing. The pair of lines in the middle run diagonally dropping from left to right. The pair run side-by-side, not crossing. The pair of lines on the right run diagonally also dropping from left to right, but with a lesser slope. The pair run side-by-side, not crossing.

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 4.27.

The figure shows two lines graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 8 to 8. The y-axis of the plane runs from negative 8 to 8. One line goes through the points (negative 5,1) and (5,5). The other line goes through the points (negative 5, negative 4) and (5,0).
Figure 4.27 Verify that both lines have the same slope, m=25m=25, 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. See Figure 4.28.

The figure shows two vertical lines graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 8 to 8. The y-axis of the plane runs from negative 8 to 8. One line goes through the points (2,1) and (2,5). The other line goes through the points (5, negative 4) and (5,0).
Figure 4.28 Vertical lines with diferent x-intercepts are parallel.

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 m1m1 and m2m2 are the slopes of two parallel lines thenm1=m2m1=m2.
  • Parallel vertical lines have different x-intercepts.

Let’s graph the equations y=−2x+3y=−2x+3 and 2x+y=−12x+y=−1 on the same grid. The first equation is already in slope–intercept form: y=−2x+3y=−2x+3. We solve the second equation for yy:

2x+y=−1y=−2x12x+y=−1y=−2x1

Graph the lines.

The figure shows two lines graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 8 to 8. The y-axis of the plane runs from negative 8 to 8. One line goes through the points (negative 4, 7) and (3, negative 7). The other line goes through the points (negative 2, 7) and (5, negative 7).

Notice the lines look parallel. What is the slope of each line? What is the y-intercept of each line?

y=mx+by=mx+by=−2x+3y=−2x1m=−2m=−2b=3,(0, 3)b=−1,(0, −1)y=mx+by=mx+by=−2x+3y=−2x1m=−2m=−2b=3,(0, 3)b=−1,(0, −1)

The slopes of the lines are the same and the y-intercept of each line is different. So we know these lines are parallel.

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.

Example 4.51

Use slopes and y-intercepts to determine if the lines 3x2y=63x2y=6 and y=32x+1y=32x+1 are parallel.

Try It 4.101

Use slopes and y-intercepts to determine if the lines 2x+5y=5andy=25x42x+5y=5andy=25x4 are parallel.

Try It 4.102

Use slopes and y-intercepts to determine if the lines 4x3y=6andy=43x14x3y=6andy=43x1 are parallel.

Example 4.52

Use slopes and y-intercepts to determine if the lines y=−4y=−4 and y=3y=3 are parallel.

Try It 4.103

Use slopes and y-intercepts to determine if the lines y=8andy=−6y=8andy=−6 are parallel.

Try It 4.104

Use slopes and y-intercepts to determine if the lines y=1andy=−5y=1andy=−5 are parallel.

Example 4.53

Use slopes and y-intercepts to determine if the lines x=−2x=−2 and x=−5x=−5 are parallel.

Try It 4.105

Use slopes and y-intercepts to determine if the lines x=1x=1 and x=−5x=−5 are parallel.

Try It 4.106

Use slopes and y-intercepts to determine if the lines x=8x=8 and x=−6x=−6 are parallel.

Example 4.54

Use slopes and y-intercepts to determine if the lines y=2x3y=2x3 and −6x+3y=−9−6x+3y=−9 are parallel. You may want to graph these lines, too, to see what they look like.

Try It 4.107

Use slopes and y-intercepts to determine if the lines y=12x1y=12x1 and x+2y=2x+2y=2 are parallel.

Try It 4.108

Use slopes and y-intercepts to determine if the lines y=34x3y=34x3 and 3x4y=123x4y=12 are parallel.

Use Slopes to Identify Perpendicular Lines

Let’s look at the lines whose equations are y=14x1y=14x1 and y=−4x+2y=−4x+2, shown in Figure 4.29.

The figure shows two lines graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 8 to 8. The y-axis of the plane runs from negative 8 to 8. One line is labeled with the equation y equals negative 4x plus 2 and goes through the points (0,2) and (1, negative 2). The other line is labeled with the equation y equals one fourth x minus 1 and goes through the points (0, negative 1) and (4,0).
Figure 4.29

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

What do you notice about the slopes of these two lines? As we read from left to right, the line y=14x1y=14x1 rises, so its slope is positive. The liney=−4x+2y=−4x+2 drops from left to right, so it has a negative slope. Does it make sense to you that the slopes of two perpendicular lines will have opposite signs?

If we look at the slope of the first line, m1=14m1=14, and the slope of the second line, m2=−4m2=−4, we can see that they are negative reciprocals of each other. If we multiply them, their product is −1.−1.

m1·m214(−4)1m1·m214(−4)1

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

Perpendicular Lines

Perpendicular lines are lines in the same plane that form a right angle.

If m1andm2m1andm2 are the slopes of two perpendicular lines, then:

m1·m2=−1andm1=−1m2m1·m2=−1andm1=−1m2

Vertical lines and horizontal lines 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 negative reciprocals. If the product of the slopes is −1−1, the lines are perpendicular. Perpendicular lines may have the same y-intercepts.

Example 4.55

Use slopes to determine if the lines, y=−5x4y=−5x4 and x5y=5x5y=5 are perpendicular.

Try It 4.109

Use slopes to determine if the lines y=−3x+2y=−3x+2 and x3y=4x3y=4 are perpendicular.

Try It 4.110

Use slopes to determine if the lines y=2x5y=2x5 and x+2y=−6x+2y=−6 are perpendicular.

Example 4.56

Use slopes to determine if the lines, 7x+2y=37x+2y=3 and 2x+7y=52x+7y=5 are perpendicular.

Try It 4.111

Use slopes to determine if the lines 5x+4y=15x+4y=1 and 4x+5y=34x+5y=3 are perpendicular.

Try It 4.112

Use slopes to determine if the lines 2x9y=32x9y=3 and 9x2y=19x2y=1 are perpendicular.

Media Access Additional Online Resources

Access this online resource for additional instruction and practice with graphs.

Section 4.5 Exercises

Practice Makes Perfect

Recognize the Relation Between the Graph and the Slope–Intercept Form of an Equation of a Line

In the following exercises, use the graph to find the slope and y-intercept of each line. Compare the values to the equation y=mx+by=mx+b.

288.
The figure shows a line graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The line goes through the points (0, negative 5) and (1, negative 2).

y=3x5y=3x5

289.
The figure shows a line graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The line goes through the points (0, negative 2) and (1,2).

y=4x2y=4x2

290.
The figure shows a line graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The line goes through the points (0,4) and (1,3).

y=x+4y=x+4

291.
The figure shows a line graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The line goes through the points (0,1) and (1, negative 2).

y=−3x+1y=−3x+1

292.
The figure shows a line graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The line goes through the points (0,1) and (3, negative 3).

y=43x+1y=43x+1

293.
The figure shows a line graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The line goes through the points (0,3) and (1,5).

y=25x+3y=25x+3

Identify the Slope and y-Intercept From an Equation of a Line

In the following exercises, identify the slope and y-intercept of each line.

294.

y=−7x+3y=−7x+3

295.

y=−9x+7y=−9x+7

296.

y=6x8y=6x8

297.

y=4x10y=4x10

298.

3x+y=53x+y=5

299.

4x+y=84x+y=8

300.

6x+4y=126x+4y=12

301.

8x+3y=128x+3y=12

302.

5x2y=65x2y=6

303.

7x3y=97x3y=9

Graph a Line Using Its Slope and Intercept

In the following exercises, graph the line of each equation using its slope and y-intercept.

304.

y=x+3y=x+3

305.

y=x+4y=x+4

306.

y=3x1y=3x1

307.

y=2x3y=2x3

308.

y=x+2y=x+2

309.

y=x+3y=x+3

310.

y=x4y=x4

311.

y=x2y=x2

312.

y=34x1y=34x1

313.

y=25x3y=25x3

314.

y=35x+2y=35x+2

315.

y=23x+1y=23x+1

316.

3x4y=83x4y=8

317.

4x3y=64x3y=6

318.

y=0.1x+15y=0.1x+15

319.

y=0.3x+25y=0.3x+25

Choose the Most Convenient Method to Graph a Line

In the following exercises, determine the most convenient method to graph each line.

320.

x=2x=2

321.

y=4y=4

322.

y=5y=5

323.

x=−3x=−3

324.

y=−3x+4y=−3x+4

325.

y=−5x+2y=−5x+2

326.

xy=5xy=5

327.

xy=1xy=1

328.

y=23x1y=23x1

329.

y=45x3y=45x3

330.

y=−3y=−3

331.

y=−1y=−1

332.

3x2y=−123x2y=−12

333.

2x5y=−102x5y=−10

334.

y=14x+3y=14x+3

335.

y=13x+5y=13x+5

Graph and Interpret Applications of Slope–Intercept

336.

The equation P=31+1.75wP=31+1.75w 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.

  1. Find Tuyet’s payment for a month when 0 units of water are used.
  2. Find Tuyet’s payment for a month when 12 units of water are used.
  3. Interpret the slope and P-intercept of the equation.
  4. Graph the equation.
337.

The equation P=28+2.54wP=28+2.54w models the relation between the amount of Randy’s monthly water bill payment, P, in dollars, and the number of units of water, w, used.

  1. Find the payment for a month when Randy used 0 units of water.
  2. Find the payment for a month when Randy used 15 units of water.
  3. Interpret the slope and P-intercept of the equation.
  4. Graph the equation.
338.

Bruce drives his car for his job. The equation R=0.575m+42R=0.575m+42 models the relation between the amount in dollars, R, that he is reimbursed and the number of miles, m, he drives in one day.

  1. Find the amount Bruce is reimbursed on a day when he drives 0 miles.
  2. Find the amount Bruce is reimbursed on a day when he drives 220 miles.
  3. Interpret the slope and R-intercept of the equation.
  4. Graph the equation.
339.

Janelle is planning to rent a car while on vacation. The equation C=0.32m+15C=0.32m+15 models the relation between the cost in dollars, C, per day and the number of miles, m, she drives in one day.

  1. Find the cost if Janelle drives the car 0 miles one day.
  2. Find the cost on a day when Janelle drives the car 400 miles.
  3. Interpret the slope and C–intercept of the equation.
  4. Graph the equation.
340.

Cherie works in retail and her weekly salary includes commission for the amount she sells. The equation S=400+0.15cS=400+0.15c models the relation between her weekly salary, S, in dollars and the amount of her sales, c, in dollars.

  1. Find Cherie’s salary for a week when her sales were 0.
  2. Find Cherie’s salary for a week when her sales were 3600.
  3. Interpret the slope and S–intercept of the equation.
  4. Graph the equation.
341.

Patel’s weekly salary includes a base pay plus commission on his sales. The equation S=750+0.09cS=750+0.09c models the relation between his weekly salary, S, in dollars and the amount of his sales, c, in dollars.

  1. Find Patel’s salary for a week when his sales were 0.
  2. Find Patel’s salary for a week when his sales were 18,540.
  3. Interpret the slope and S-intercept of the equation.
  4. Graph the equation.
342.

Costa is planning a lunch banquet. The equation C=450+28gC=450+28g models the relation between the cost in dollars, C, of the banquet and the number of guests, g.

  1. Find the cost if the number of guests is 40.
  2. Find the cost if the number of guests is 80.
  3. Interpret the slope and C-intercept of the equation.
  4. Graph the equation.
343.

Margie is planning a dinner banquet. The equation C=750+42gC=750+42g models the relation between the cost in dollars, C of the banquet and the number of guests, g.

  1. Find the cost if the number of guests is 50.
  2. Find the cost if the number of guests is 100.
  3. Interpret the slope and C–intercept of the equation.
  4. Graph the equation.

Use Slopes to Identify Parallel Lines

In the following exercises, use slopes and y-intercepts to determine if the lines are parallel.

344.

y=34x3;3x4y=2y=34x3;3x4y=2

345.

y=23x1;2x3y=2y=23x1;2x3y=2

346.

2x5y=3;y=25x+12x5y=3;y=25x+1

347.

3x4y=2;y=34x33x4y=2;y=34x3

348.

2x4y=6;x2y=32x4y=6;x2y=3

349.

6x3y=9;2xy=36x3y=9;2xy=3

350.

4x+2y=6;6x+3y=34x+2y=6;6x+3y=3

351.

8x+6y=6;12x+9y=128x+6y=6;12x+9y=12

352.

x=5;x=6x=5;x=6

353.

x=7;x=8x=7;x=8

354.

x=4;x=1x=4;x=1

355.

x=3;x=2x=3;x=2

356.

y=2;y=6y=2;y=6

357.

y=5;y=1y=5;y=1

358.

y=4;y=3y=4;y=3

359.

y=1;y=2y=1;y=2

360.

xy=2;2x2y=4xy=2;2x2y=4

361.

4x+4y=8;x+y=24x+4y=8;x+y=2

362.

x3y=6;2x6y=12x3y=6;2x6y=12

363.

5x2y=11;5xy=75x2y=11;5xy=7

364.

3x6y=12;6x3y=33x6y=12;6x3y=3

365.

4x8y=16;x2y=44x8y=16;x2y=4

366.

9x3y=6;3xy=29x3y=6;3xy=2

367.

x5y=10;5xy=10x5y=10;5xy=10

368.

7x4y=8;4x+7y=147x4y=8;4x+7y=14

369.

9x5y=4;5x+9y=19x5y=4;5x+9y=1

Use Slopes to Identify Perpendicular Lines

In the following exercises, use slopes and y-intercepts to determine if the lines are perpendicular.

370.

3x2y=8;2x+3y=63x2y=8;2x+3y=6

371.

x4y=8;4x+y=2x4y=8;4x+y=2

372.

2x+5y=3;5x2y=62x+5y=3;5x2y=6

373.

2x+3y=5;3x2y=72x+3y=5;3x2y=7

374.

3x2y=1;2x3y=23x2y=1;2x3y=2

375.

3x4y=8;4x3y=63x4y=8;4x3y=6

376.

5x+2y=6;2x+5y=85x+2y=6;2x+5y=8

377.

2x+4y=3;6x+3y=22x+4y=3;6x+3y=2

378.

4x2y=5;3x+6y=84x2y=5;3x+6y=8

379.

2x6y=4;12x+4y=92x6y=4;12x+4y=9

380.

6x4y=5;8x+12y=36x4y=5;8x+12y=3

381.

8x2y=7;3x+12y=98x2y=7;3x+12y=9

Everyday Math

382.

The equation C=59F17.8C=59F17.8 can be used to convert temperatures F, on the Fahrenheit scale to temperatures, C, on the Celsius scale.

  1. Explain what the slope of the equation means.
  2. Explain what the C–intercept of the equation means.
383.

The equation n=4T160n=4T160 is used to estimate the number of cricket chirps, n, in one minute based on the temperature in degrees Fahrenheit, T.

  1. Explain what the slope of the equation means.
  2. Explain what the n–intercept of the equation means. Is this a realistic situation?

Writing Exercises

384.

Explain in your own words how to decide which method to use to graph a line.

385.

Why are all horizontal lines parallel?

Self Check

After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

This table has eight rows and four columns. The first row is a header row and it labels each column. The first column is labeled "I can …", the second "Confidently", the third “With some help” and the last "No–I don’t get it". In the “I can…” column the next row reads “recognize the relation between the graph and the slope-intercept form of an equation of a line.” The third row reads “identify the Slope and y-intercept from an equation of a line”. The fourth row reads “graph a line using its slope and intercept”. The fifth row reads “choose the most convenient method to graph a line.” The sixth row reads “graph and interpret applications of slope-intercept”. The seventh row reads “use slopes to identify parallel lines” and the last row reads “use slopes to identify perpendicular lines.” The remaining columns are blank.

After looking at the checklist, do you think you are well-prepared for the next section? Why or why not?

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