4.1 The Graph of a Linear Equation - McGraw Hill · PDF file4.1 The Graph of a Linear Equation 4.1 OBJECTIVES 1. Find three ordered pairs for an equation in two ... in which A and

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The Graph of a Linear Equation4.1

4.1 OBJECTIVES

1. Find three ordered pairs for an equation in twovariables

2. Graph a line from three points3. Graph a line by the intercept method4. Graph a line that passes through the origin5. Determine domain and range 6. Graph horizontal and vertical lines

In previous algebra classes you have solved equations in one variable such as 3x � 2 � 5x � 4.

Solving such an equation required finding the value of the variable, in this case x, thatmade the equation a true statement. In this case, that value is x � �3, because

3(�3) � 2 � 5(�3) � 4

This is a true statement because each side of the equation is equal to �11; no other valuefor x makes this statement true. The solution can be written in three different ways. We canwrite x � �3, �x�x � �3� which is read “the set of all x such that x equals �3,” or simply��3�, which is the set containing the number �3.

What if we have an equation in two variables, such as 3x � y � 6? The solution set isdefined in a similar manner.

The solution set for an equation in two variables is a set of ordered pairs. Typically, therewill be an infinite number of ordered pairs that make an equation a true statement. We canfind some of these ordered pairs by substituting a value for x, then solving the remainingequation for y. We will use that technique in Example 1.

The solution set for an equation in two variables is the set containing all orderedpairs of real numbers (x, y) that will make the equation a true statement.

Definitions: Solution Set for an Equation in Two Variables

Finding Ordered Pair Solutions

Find three ordered pairs that are solutions for each equation.

(a) 3x � y � 6

We will pick three values for x, set up a table for ordered pairs, and then determine therelated value for y.

x y

�101

Example 1

THE GRAPH OF A LINEAR EQUATION SECTION 4.1 191©

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Substituting �1 for x, we get

3(�1) � y � 6

�3 � y � 6

y � 9

The ordered pair (�1, 9) is a solution to the equation 3x � y � 6.Substituting 0 for x, we get

3(0) � y � 6

0 � y � 6

y � 6

The ordered pair (0, 6) is a solution to the equation 3x � y � 6.Substituting 1 for x, we get

3(1) � y � 6

3 � y � 6

y � 3

The ordered pair (1, 3) is a solution to the equation 3x � y � 6.Completing the table gives us the following:

(b) 2x � y � 1

Let’s try a different set of values for x. We will use the following table.

Substituting �5 for x, we get

2(�5) � y � 1

�10 � y � 1

�y � 11

y � �11

The ordered pair (�5, �11) is a solution to the equation 2x � y � 1.Substituting 0 for x, we get

2(0) � y � 1

0 � y � 1

�y � 1

y � �1

x y

�505

x y

�1 90 61 3

NOTE To indicate the set of allsolutions to the equation, wewrite

{(x, y)�3x � y � 6}

The ordered pair (0, �1) is a solution to the equation 2x � y � 1.Substituting 5 for x, we get

2(5) � y � 1

10 � y � 1

�y � �9

y � 9

The ordered pair (5, 9) is a solution to the equation 2x � y � 1.Completing the table gives us the following:

x y

�5 �110 �15 9

192 CHAPTER 4 GRAPHS OF LINEAR EQUATIONS AND FUNCTIONS

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C H E C K Y O U R S E L F 1

Find three ordered pairs that are solutions for each equation.

(a) 2x � y � 6 (b) 3x � y � 2

The graph of the solution set of an equation in two variables, usually called the graphof the equation, is the set of all points with coordinates (x, y) that satisfy the equation.

In this chapter, we are primarily interested in a particular kind of equation in x and y andthe graph of that equation. The equations we refer to involve x and y to the first power, andthey are called linear equations.

NOTE Why can A and B notboth be zero? First, recall that,although x and y are variables,A, B, and C are constants. Withthat in mind, look at theequation if A and B are bothzero.

(0)x � (0)y � C

0 � 0 � C

0 � C

Because zero must be aconstant, we are left with thestatement

0 � 0

This would be a true statementregardless of the values of xand y. Its graph would be everypoint in the plane.

Graphing by Plotting Points

Graph the equation

x � y � 5

This is a linear equation in two variables. To draw its graph, we can begin by assigningvalues to x and finding the corresponding values for y. For instance, if x � 1, we have

1 � y � 5

y � 4

Therefore, (1, 4) satisfies the equation and is on the graph of x � y � 5.

An equation of the form

Ax � By � C

in which A and B cannot both be zero, is called the standard form for a line. Itsgraph is always a line.

Definitions: Linear Equations

Example 2

NOTE Because two pointsdetermine a line, technicallytwo points are all that areneeded to graph the equation.You may want to locate at leastone other point as a check ofyour work.

NOTE Again, the set of allsolutions is

{(x, y)�2x � y � 1}


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NOTE If you first rewrite anequation so that y is isolated onthe left side, it can be easilyentered and graphed with agraphing calculator. In this case,graph the equation

y � �x � 5

Similarly, (2, 3), (3, 2), and (4, 1) are in the graph. Often these results are recorded in atable of values, as shown below. We then plot the points determined and draw a line throughthose points.

x � y � 5

Every point on the graph of the equation x � y � 5 has coordinates that satisfy theequation, and every point with coordinates that satisfy the equation lies on the line.

(1, 4) (2, 3)

(3, 2) (4, 1)

y

x

x y

1 42 33 24 1

Graphing by the Intercept Method

Use the intercepts to graph the equation

x � 2y � 6

To find the x intercept, let y � 0.

x � 2 � 0 � 6

x � 6

The x intercept is (6, 0).


Graph the equation 2x � y � 6.

The following algorithm summarizes our first approach to graphing a linear equationin two variables.

Two particular points are often used in graphing an equation because they are very easyto find. The x intercept of a line is the point at which the line crosses the x axis. If the xintercept exists, it can be found by setting y � 0 in the equation and solving for x. The yintercept is the point at which the line crosses the y axis. If the y intercept exists, it is foundby letting x � 0 and solving for y.

Step 1 Find at least three solutions for the equation, and write your results ina table of values.

Step 2 Graph the points associated with the ordered pairs found in step 1.Step 3 Draw a line through the points plotted above to form the graph of the

equation.

Step by Step: To Graph a Linear Equation

NOTE An algorithm is asequence of steps that solve aproblem.

Example 3

NOTE Solving for y, we get

y � x � 3

To graph this result on yourcalculator, you can enter

Y1 � (1 � 2)x � 3

using the key for x.x, T, u, n

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Step 1 Find the x intercept. Let y � 0, and solve for x.Step 2 Find the y intercept. Let x � 0, and solve for y.Step 3 Plot the two intercepts determined in steps 1 and 2.Step 4 Draw a line through the intercepts.

y intercept(x � 0)

x intercept(y � 0)

y

x

Step by Step: Graphing by the Intercept Method


Graph, using the intercept method.

4x � 3y � 12

When can the intercept method not be used? Some lines have only one intercept. Forinstance, the graph of x � 2y � 0 passes through the origin. In this case, other points mustbe used to graph the equation.

The following algorithm summarizes the steps of graphing a line by the intercept method.

To find the y intercept, let x � 0.

0 � 2y � 6

�2y � 6

y � �3

The y intercept is (0, �3).Graphing the intercepts and drawing the line through those intercepts, we have the de-

sired graph.

(6, 0)

(0, �3)

y

x � 2y � 6

x


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Graphing a Line That Passes Through the Origin

Graph x � 2y � 0.Letting y � 0 gives

x � 2 � 0 � 0

x � 0

Thus (0, 0) is a solution, and the line has only one intercept.We continue by choosing any other convenient values for x. If x � 2:

2 � 2y � 0

2y � �2

y � �1

So (2, �1) is a solution. You can easily verify that (4, �2) is also a solution. Again, plot-ting the points and drawing the line through those points, we have the desired graph.

y

x(2, �1)

(4, �2)

x � 2y � 0

(0, 0)

Example 4

NOTE Graph the equation

y � � x

Note that the line passesthrough the origin.

12


Graph the equation x � 3y � 0.

In Section 3.1, we defined the terms domain and range. Recall that the domain of a re-lation is the set of all the first elements in the ordered pairs. The range is the set of all thesecond elements. Recall that a line is the graph of a set of ordered pairs. In Example 5, wewill examine the domain and range for the graph of a line.

Finding the Domain and Range

Find the domain and range for the relation described by the equation

x � y � 5

We can analyze the domain and range either graphically or algebraically. First, we willlook at a graphical analysis. From Example 2, let’s look at the graph of the equation.

(1, 4) (2, 3)

(3, 2) (4, 1)

y

x

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Find the domain and range for the relation described by the following equation.

x � y � 4

1. A line with an equation of the form

y � k

is horizontal (parallel to the x axis).2. A line with an equation of the form

x � h

is vertical (parallel to the y axis).

Rules and Properties: Vertical or Horizontal Lines

Two types of linear equations are worthy of special attention. Their graphs are lines thatare parallel to the x or y axis, and the equations are special cases of the general form

Ax � By � C

in which either A � 0 or B � 0.

The graph continues forever at both ends. For every value of x, there is an associated pointon the line. Therefore, the domain (D) is the set of all real numbers. In set notation, we write

D � �x �x � R�

This is read, “The domain is the set of every x that is a real number.”To find the range (R), we look at the graph to see what values are associated with y. Note

that every y is associated with some point. The range is written as

R � �y �y � R�

This is read, “The range is the set of every y that is a real number.”Let’s find the domain and range for the same relation by using an algebraic analysis.

Look at the following equation.

x � y � 5

To determine the domain, we need to find every value of x that allows us to solve for y. Thatcombination will result in an ordered pair (x, y). The set of all those x values is the domainof the relation.

We can find a value for y for any real value of x. For example, if x � �5,

�5 � y � 5

y � 10

The ordered pair (�5, 10) is part of the relation. As in our graphical analysis, the domainis

D � �x �x � R�

By a similar argument, we can substitute any value for y and solve the equation for x. Therange is

R � �y �y � R�


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Example 6 illustrates both cases.

Graphing Horizontal and Vertical Lines

(a) Graph the line with equation

y � 3

You can think of the equation in the equivalent form

0 � x � y � 3

Note that any ordered pair of the form (__, 3) will satisfy the equation. Because x is multi-plied by 0, y will always be equal to 3.

For instance, (�2, 3) and (3, 3) are on the graph. The graph, a horizontal line, is shownbelow.

The domain for a horizontal line is every real number. The range is a single y value. Wewrite

D � �x �x � R� and R � �3�

(b) Graph the line with equation

x � �2

In this case, you can think of the equation in the equivalent form

x � 0 � y � �2

Now any ordered pair of the form (�2, __) will satisfy the equation. Examples are (�2, �1)and (�2, 3). The graph, a vertical line, is shown below.

y

x

x � �2

y

x

y � 3

Example 6

NOTE Because part (a) is afunction, it can be graphed onyour calculator. Part (b) is not afunction and cannot begraphed on your calculator.

NOTE Notice that

D � ��2�

and

R � �y �y � R�


Graph each equation and state the domain and range.

(a) y � �3 (b) x � 5


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C H E C K Y O U R S E L F A N S W E R S

1. (a) Answers will vary, but could include (0, �6); (b) Answers will vary, but could

include (0, 2). 2.

3. 4. 5. D � �x �x � R� and

R � �y � y � R�

6. (a) (b)

D � �x �x � R� D � �5�R � ��3� R � �y � y � R�

y

x

x � 5

(5, 0)

y

x

y � �3(0, �3)

y

x(0, 0)

(3, 1)(6, 2)

x � 3y � 0

(0, 4)

(3, 0)

y

x

4x � 3y � 12

y

x

(2, �2)

(1, �4)

(0, �6)

x y

0 �61 �42 �2

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4.1

Name

Section Date

ANSWERS

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

16.

17.

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Exercises

In exercises 1 to 8, find three ordered pairs that are solutions to the given equations.

1. 2x � y � 5 2. 3x � y � 7 3. 7x � y � 8 4. 5x � y � 3

5. 4x � 5y � 20 6. 2x � 3y � 6 7. 3x � y � 0 8. 2x � y � 0

In exercises 9 to 26, graph each of the equations.

9. x � y � 6 10. x � y � 6 11. y � x � 2

12. y � x � 5 13. y � x � 1 14. y � 2x � 2

15. y � �2x � 1 16. y � �3x � 1 17. y � x � 3

y

x

y

x

y

x

1

2

y

x

y

x

y

x

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18. y � 2x � 4 19. y � �x � 3 20. y � �2x � 4

21. x � 2y � 0 22. x � 2y � 0 23. x � 4

24. x � �4 25. y � 4 26. y � �6

y

x

y

x

y

x

y

x

y

x

y

x

y

x

y

x

y

x

ANSWERS

18.

19.

20.

21.

22.

23.

24.

25.

26.

200

In exercises 27 to 38, find the x and y intercepts and then graph each equation.

27. x � 2y � 4 28. x � 3y � 6 29. 2x � y � 6

30. 3x � 2y � 12 31. 2x � 5y � 10 32. 2x � 3y � 6

33. 5x � 6y � 0 34. 2x � 7y � 0 35. x � 4y � 8 � 0

y

x

y

x

y

x

y

x

y

x

y

x

y

x

y

x

y

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27.

28.

29.

30.

31.

32.

33.

34.

35.

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36. 2x � y � 6 � 0 37. 8x � 4y 38. 6x � �7y

In exercises 39 to 46, find the domain and range of each of the relations.

39. 3x � 2y � 4 40. 5x � 4y � 20

41. 6x � 2y � 18 42. �x � 5y � 8

43. x � 4 44. 2x � 10 � 0

45. y � 3 46. 3y � 12 � 0

For exercises 47 to 54, select a window that allows you to see both the x and y interceptson your calculator. If that is not possible, explain why not.

47. x � y � 40 48. x � y � 80

49. 2x � 3y � 900 50. 5x � 8y � 800

51. y � 5x � 90 52. y � 3x � 450

53. y � 30x 54. y � 200

y

x

y

x

y

x

ANSWERS

36.

37.

38.

39.

40.

41.

42.

43.

44.

45.

46.

47.

48.

49.

50.

51.

52.

53.

54.

202

Two distinct lines in the plane either are parallel or they intersect. In exercises 55 to 58,graph each pair of equations on the same set of axes, and find the point of intersection,where possible.

55. x � y � 6, x � y � 4 56. y � x � 3, y � �x � 1

57. y � 2x, y � x � 1 58. 2x � y � 3, 2x � y � 5

59. Graph y � x and y � 2x on the same set of axes. What do you observe?

60. Graph y � 2x � 1 and y � �2x � 1 on the same set of axes. What doyou observe?

61. Graph y � 2x and y � 2x � 1 on the same set of axes. What do you observe?

62. Graph y � 3x � 1 and y � 3x � 1 on the same set of axes. What do you observe?

63. Graph y � 2x and y � � x on the same set of axes. What do you observe?

64. Graph y � x � and y � �3x � 2 on the same set of axes. What do

you observe?

7

3

1

3

1

2

y

x

y

x

y

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55.

56.

57.

58.

59.

60.

61.

62.

63.

64.

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Use your graphing utility to graph each of the following equations.

65. y � �3 66. y � 2

67. y � 3x � 1 68. y � �2x � 2

69. Write an equation whose graph will have no x intercept but will have a y intercept at (0,6).

70. Write an equation whose graph will have no y intercept but will have an x interceptat (�5, 0).

Answers1. (0, 5), (1, 3), (�1, 7) 3. (0, �8), (1, �1), (�1, �15)

5. (0, 4), (5, 0), ��1, � 7. (0, 0), (1, �3), (�1, 3)

9. x � y � 6 11. y � x � 2 13. y � x � 1

15. y � �2x � 1 17. y � x � 3 19. y � �x � 3

21. x � 2y � 0 23. x � 4 25. y � 4

y

x

y

x

y

x

y

x

y

x

y

x

1

2

y

x

y

x

y

x

24

5

ANSWERS

65.

66.

67.

68.

69.

70.

204

27. x � 2y � 4; y intercept (0, �2); 29. 2x � y � 6; y intercept (0, �6);x intercept (4, 0) x intercept (3, 0)

31. 2x � 5y � 10; y intercept (0, 2); 33. 5x � 6y � 0;x intercept (5, 0) intercepts: (0, 0)

35. x � 4y � 8 � 0; y intercept (0, �2); 37. 8x � 4y;x intercept (�8, 0) intercepts: (0, 0)

39. D: �x �x � R�; R: �y �y � R�41. D: �x �x � R�; R: �y �y � R�43. D: �4�; R: �y �y � R�45. D: �x �x � R�; R: �3�47. X max � 40, Y max � 4049. X max � 450, Y max � 30051. X min � �18, Y max � 9053. Any viewing window that shows the origin

y

x

y

x

y

x

y

x

y

x

y

x

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55. Intersection: (5, 1) 57. Intersection: (1, 2)

59. The line corresponding to 61. The two lines appear to be parallel.y � 2x is steeper than thatcorresponding to y � x.

63. The lines appear to be 65.perpendicular.

67.

69. y � 6

y � 3x�1

�2��4 2 4

y

x

4

2

�2

�4

�2�4 2 4

y � �3

y

x

y

x

y

x

y

x

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Documents

4.1 The Graph of a Linear Equation - McGraw Hill · PDF file4.1 The Graph of a Linear Equation 4.1 OBJECTIVES 1. Find three ordered pairs for an equation in two ... in which A and