11.3: Graphing Linear Equations (2024)

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    Learning Objectives

    By the end of this section, you will be able to:

    • Recognize the relation between the solutions of an equation and its graph
    • Graph a linear equation by plotting points
    • Graph vertical and horizontal lines

    Be Prepared 11.4

    Before you get started, take this readiness quiz.

    Evaluate: 3x+23x+2 when x=−1.x=−1.
    If you missed this problem, review Example 3.56.

    Be Prepared 11.5

    Solve the formula: 5x+2y=205x+2y=20 for y.y.
    If you missed this problem, review Example 9.62.

    Be Prepared 11.6

    Simplify: 38(−24).38(−24).
    If you missed this problem, review Example 4.28.

    Recognize the Relation Between the Solutions of an Equation and its Graph

    In Use the Rectangular Coordinate System, we found a few solutions to the equation 3x+2y=63x+2y=6. They are listed in the table below. So, the ordered pairs (0,3)(0,3), (2,0)(2,0), (1,32)(1,32), (4,3)(4,3), are some solutions to the equation3x+2y=63x+2y=6. We can plot these solutions in the rectangular coordinate system as shown on the graph at right.

    11.3: Graphing Linear Equations (2)

    Notice how the points line up perfectly? We connect the points with a straight line to get the graph of the equation 3x+2y=63x+2y=6. Notice the arrows on the ends of each side of the line. These arrows indicate the line continues.

    11.3: Graphing Linear Equations (3)

    Every point on the line is a solution of the equation. Also, every solution of this equation is a point on this line. Points not on the line are not solutions!

    Notice that the point whose coordinates are (2,6)Figure 11.8. If you substitute x=2x=2 and y=6y=6 into the equation, you find that it is a solution to the equation.

    11.3: Graphing Linear Equations (4)

    So (4,1)(4,1) is not a solution to the equation 3x+2y=63x+2y=6 . Therefore the point (4,1)(4,1) is not on the line.

    This is an example of the saying,” A picture is worth a thousand words.” The line shows you all the solutions to the equation. Every point on the line is a solution of the equation. And, every solution of this equation is on this line. This line is called the graph of the equation 3x+2y=63x+2y=6.

    Graph of a Linear Equation

    The graph of a linear equation Ax+By=CAx+By=C is a straight line.

    • Every point on the line is a solution of the equation.
    • Every solution of this equation is a point on this line.

    Example 11.15

    The graph of y=2x3y=2x3 is shown below.

    11.3: Graphing Linear Equations (5)

    For each ordered pair decide

    1. Is the ordered pair a solution to the equation?
    2. Is the point on the line?
    1. (a) (0,3)(0,3)
    2. (b) (3,3)(3,3)
    3. (c) (2,3)(2,3)
    4. (d) (1,5)(1,5)
    Answer

    Substitute the xx- and yy-values into the equation to check if the ordered pair is a solution to the equation.

    11.3: Graphing Linear Equations (6)

    Plot the points A: (0,3)(0,3) B: (3,3)(3,3) C: (2,3)(2,3) and D: (1,5)(1,5).
    The points (0,3)(0,3), (3,3)(3,3), and (1,5)(1,5) are on the line y=2x3y=2x3, and the point (2,3)(2,3) is not on the line.

    11.3: Graphing Linear Equations (7)

    The points which are solutions to y=2x3y=2x3 are on the line, but the point which is not a solution is not on the line.

    Try It 11.29

    The graph of y=3x1y=3x1 is shown.

    For each ordered pair, decide

    1. is the ordered pair a solution to the equation?
    2. is the point on the line?
    11.3: Graphing Linear Equations (8)
    1. (0,1)(0,1)
    2. (2,2)(2,2)
    3. (3,1)(3,1)
    4. (1,4)(1,4)

    Graph a Linear Equation by Plotting Points

    There are several methods that can be used to graph a linear equation. The method we used at the start of this section to graph is called plotting points, or the Point-Plotting Method.

    Let’s graph the equation y=2x+1y=2x+1 by plotting points.

    We start by finding three points that are solutions to the equation. We can choose any value for xx or y,y, and then solve for the other variable.

    Since yy is isolated on the left side of the equation, it is easier to choose values for x.x. We will use 0,1,0,1, and -2-2 for xx for this example. We substitute each value of xx into the equation and solve for y.y.

    11.3: Graphing Linear Equations (9)

    We can organize the solutions in a table. See Table 11.2.

    y=2x+1y=2x+1
    xx yy (x,y)(x,y)
    00 11 (0,1)(0,1)
    11 33 (1,3)(1,3)
    −2−2 −3−3 (−2,−3)(−2,−3)

    Table 11.2

    Now we plot the points on a rectangular coordinate system. Check that the points line up. If they did not line up, it would mean we made a mistake and should double-check all our work. See Figure 11.9.

    11.3: Graphing Linear Equations (10)

    Draw the line through the three points. Extend the line to fill the grid and put arrows on both ends of the line. The line is the graph of y=2x+1.y=2x+1.

    11.3: Graphing Linear Equations (11)

    How To

    Graph a linear equation by plotting points.

    1. Step 1. Find three points whose coordinates are solutions to the equation. Organize them in a table.
    2. Step 2. Plot the points on a rectangular coordinate system. Check that the points line up. If they do not, carefully check your work.
    3. Step 3. Draw the line through the points. Extend the line to fill the grid and put arrows on both ends of the line.

    It is true that it only takes two points to determine a line, but it is a good habit to use three points. If you plot only two points and one of them is incorrect, you can still draw a line but it will not represent the solutions to the equation. It will be the wrong line. If you use three points, and one is incorrect, the points will not line up. This tells you something is wrong and you need to check your work. See Figure 11.11.

    11.3: Graphing Linear Equations (12)

    Example 11.16

    Graph the equation y=−3x.y=−3x.

    Answer

    Find three points that are solutions to the equation. It’s easier to choose values for x,x, and solve for y.y. Do you see why?

    11.3: Graphing Linear Equations (13)

    List the points in a table.

    y=−3xy=−3x
    xx yy (x,y)(x,y)
    00 00 (0,0)(0,0)
    11 33 (1,−3)(1,−3)
    −2−2 66 (−2,6)(−2,6)

    Plot the points, check that they line up, and draw the line as shown.

    11.3: Graphing Linear Equations (14)

    Try It 11.30

    Graph the equation by plotting points: y=−4x.y=−4x.

    Try It 11.31

    Graph the equation by plotting points: y=x.y=x.

    When an equation includes a fraction as the coefficient of x,x, we can substitute any numbers for x.x. But the math is easier if we make ‘good’ choices for the values of x.x. This way we will avoid fraction answers, which are hard to graph precisely.

    Example 11.17

    Graph the equation y=12x+3.y=12x+3.

    Answer

    Find three points that are solutions to the equation. Since this equation has the fraction 1212 as a coefficient of x,x, we will choose values of xx carefully. We will use zero as one choice and multiples of 22 for the other choices.

    11.3: Graphing Linear Equations (15)

    The points are shown in the table.

    y=12x+3y=12x+3
    xx yy (x,y)(x,y)
    00 33 (0,3)(0,3)
    22 44 (2,4)(2,4)
    44 55 (4,5)(4,5)

    Plot the points, check that they line up, and draw the line as shown.

    11.3: Graphing Linear Equations (16)

    Try It 11.32

    Graph the equation: y=13x1.y=13x1.

    Try It 11.33

    Graph the equation: y=14x+2.y=14x+2.

    So far, all the equations we graphed had yy given in terms of x.x. Now we’ll graph an equation with xx and yy on the same side.

    Example 11.18

    Graph the equation x+y=5.x+y=5.

    Answer

    Find three points that are solutions to the equation. Remember, you can start with any value of xx or y.y.

    11.3: Graphing Linear Equations (17)

    We list the points in a table.

    x+y=5x+y=5
    xx yy (x,y)(x,y)
    00 55 (0,5)(0,5)
    11 44 (1,4)(1,4)
    44 11 (4,1)(4,1)

    Then plot the points, check that they line up, and draw the line.

    11.3: Graphing Linear Equations (18)

    Try It 11.34

    Graph the equation: x+y=−2.x+y=−2.

    Try It 11.35

    Graph the equation: xy=6.xy=6.

    In the previous example, the three points we found were easy to graph. But this is not always the case. Let’s see what happens in the equation 2x+y=3.2x+y=3. If yy is 0,0, what is the value of x?x?

    11.3: Graphing Linear Equations (19)

    The solution is the point (32,0).(32,0). This point has a fraction for the xx-coordinate. While we could graph this point, it is hard to be precise graphing fractions. Remember in the example y=12x+3,y=12x+3, we carefully chose values for xx so as not to graph fractions at all. If we solve the equation 2x+y=32x+y=3 for y,y, it will be easier to find three solutions to the equation.

    2x+y=32x+y=3

    y=−2x+3y=−2x+3

    Now we can choose values for xx that will give coordinates that are integers. The solutions for x=0,x=1,x=0,x=1, and x=−1x=−1 are shown.

    y=−2x+3y=−2x+3
    xx yy (x,y)(x,y)
    00 33 (0,3)(0,3)
    11 11 (1,1)(1,1)
    −1−1 55 (-1,5)(-1,5)

    11.3: Graphing Linear Equations (20)

    Example 11.19

    Graph the equation 3x+y=−1.3x+y=−1.

    Answer

    Find three points that are solutions to the equation.

    First, solve the equation for y.y.

    3x+y=−1 y=−3x1 3x+y=−1 y=−3x1

    We’ll let xx be 0,1,0,1, and −1−1 to find three points. The ordered pairs are shown in the table. Plot the points, check that they line up, and draw the line.

    y=−3x1y=−3x1
    xx yy (x,y)(x,y)
    00 −1−1 (0,−1)(0,−1)
    11 −4−4 (1,−4)(1,−4)
    −1−1 22 (−1,2)(−1,2)

    11.3: Graphing Linear Equations (21)

    If you can choose any three points to graph a line, how will you know if your graph matches the one shown in the answers in the book? If the points where the graphs cross the x-x- and yy-axes are the same, the graphs match.

    Try It 11.36

    Graph each equation: 2x+y=2.2x+y=2.

    Try It 11.37

    Graph each equation: 4x+y=−3.4x+y=−3.

    Graph Vertical and Horizontal Lines

    Can we graph an equation with only one variable? Just xx and no y,y, or just yy without an x?x? How will we make a table of values to get the points to plot?

    Let’s consider the equation x=−3.x=−3. The equation says that xx is always equal to −3,−3, so its value does not depend on y.y. No matter what yy is, the value of xx is always −3.−3.

    To make a table of solutions, we write −3−3 for all the xx values. Then choose any values for y.y. Since xx does not depend on y,y, you can chose any numbers you like. But to fit the size of our coordinate graph, we’ll use 1,2,1,2, and 33 for the yy-coordinates as shown in the table.

    x=−3x=−3
    xx yy (x,y)(x,y)
    −3−3 11 (−3,1)(−3,1)
    −3−3 22 (−3,2)(−3,2)
    −3−3 33 (−3,3)(−3,3)

    Then plot the points and connect them with a straight line. Notice in Figure 11.12 that the graph is a vertical line.

    11.3: Graphing Linear Equations (22)

    Vertical Line

    A vertical line is the graph of an equation that can be written in the form x=a.x=a.

    The line passes through the xx-axis at (a,0)(a,0).

    Example 11.20

    Graph the equation x=2.x=2. What type of line does it form?

    Answer

    The equation has only variable, x,x, and xx is always equal to 2.2. We make a table where xx is always 22 and we put in any values for y.y.

    x=2x=2
    xx yy (x,y)(x,y)
    22 11 (2,1)(2,1)
    22 22 (2,2)(2,2)
    22 33 (2,3)(2,3)

    Plot the points and connect them as shown.

    11.3: Graphing Linear Equations (23)

    The graph is a vertical line passing through the xx-axis at 2.2.

    Try It 11.38

    Graph the equation: x=5.x=5.

    Try It 11.39

    Graph the equation: x=−2.x=−2.

    What if the equation has yy but no xx? Let’s graph the equation y=4.y=4. This time the yy-value is a constant, so in this equation yy does not depend on x.x.

    To make a table of solutions, write 44 for all the yy values and then choose any values for x.x.

    We’ll use 0,2,0,2, and 44 for the xx-values.

    y=4y=4
    xx yy (x,y)(x,y)
    00 44 (0,4)(0,4)
    22 44 (2,4)(2,4)
    44 44 (4,4)(4,4)

    Plot the points and connect them, as shown in Figure 11.13. This graph is a horizontal line passing through the y-axisy-axis at 4.4.

    11.3: Graphing Linear Equations (24)

    Horizontal Line

    A horizontal line is the graph of an equation that can be written in the form y=b.y=b.

    The line passes through the y-axisy-axis at (0,b).(0,b).

    Example 11.21

    Graph the equation y=−1.y=−1.

    Answer

    The equation y=−1y=−1 has only variable, y.y. The value of yy is constant. All the ordered pairs in the table have the same yy-coordinate, −1−1. We choose 0,3,0,3, and −3−3 as values for x.x.

    y=−1y=−1
    xx yy (x,y)(x,y)
    −3−3 −1−1 (−3,−1)(−3,−1)
    00 −1−1 (0,−1)(0,−1)
    33 −1−1 (3,−1)(3,−1)

    The graph is a horizontal line passing through the yy-axis at –1–1 as shown.

    11.3: Graphing Linear Equations (25)

    Try It 11.40

    Graph the equation: y=−4.y=−4.

    Try It 11.41

    Graph the equation: y=3.y=3.

    The equations for vertical and horizontal lines look very similar to equations like y=4x.y=4x. What is the difference between the equations y=4xy=4x and y=4?y=4?

    The equation y=4xy=4x has both xx and y.y. The value of yy depends on the value of x.x. The y-coordinatey-coordinate changes according to the value of x.x.

    The equation y=4y=4 has only one variable. The value of yy is constant. The y-coordinatey-coordinate is always 4.4. It does not depend on the value of x.x.

    11.3: Graphing Linear Equations (26)

    The graph shows both equations.

    11.3: Graphing Linear Equations (27)

    Notice that the equation y=4xy=4x gives a slanted line whereas y=4y=4 gives a horizontal line.

    Example 11.22

    Graph y=−3xy=−3x and y=−3y=−3 in the same rectangular coordinate system.

    Answer

    Find three solutions for each equation. Notice that the first equation has the variable x,x, while the second does not. Solutions for both equations are listed.

    11.3: Graphing Linear Equations (28)

    The graph shows both equations.

    11.3: Graphing Linear Equations (29)

    Try It 11.42

    Graph the equations in the same rectangular coordinate system: y=−4xy=−4x and y=−4.y=−4.

    Try It 11.43

    Graph the equations in the same rectangular coordinate system: y=3y=3 and y=3x.y=3x.

    Media

    Section 11.2 Exercises

    Practice Makes Perfect

    Recognize the Relation Between the Solutions of an Equation and its Graph

    In each of the following exercises, an equation and its graph is shown. For each ordered pair, decide

    1. is the ordered pair a solution to the equation?
    2. is the point on the line?

    39.

    y=x+2y=x+2

    11.3: Graphing Linear Equations (30)

    1. (0,2)(0,2)
    2. (1,2)(1,2)
    3. (1,1)(1,1)
    4. ( 3 , 1 ) ( 3 , 1 )

    40.

    y=x4y=x4

    11.3: Graphing Linear Equations (31)

    1. (0,4)(0,4)
    2. (3,1)(3,1)
    3. (2,2)(2,2)
    4. ( 1 , 5 ) ( 1 , 5 )

    41.

    y=12x3y=12x3

    11.3: Graphing Linear Equations (32)

    1. (0,3)(0,3)
    2. (2,2)(2,2)
    3. (2,4)(2,4)
    4. ( 4 , 1 ) ( 4 , 1 )

    42.

    y=13x+2y=13x+2

    11.3: Graphing Linear Equations (33)

    1. (0,2)(0,2)
    2. (3,3)(3,3)
    3. (3,2)(3,2)
    4. ( 6 , 0 ) ( 6 , 0 )

    Graph a Linear Equation by Plotting Points

    In the following exercises, graph by plotting points.

    43.

    y = 3 x 1 y = 3 x 1

    44.

    y = 2 x + 3 y = 2 x + 3

    45.

    y = −2 x + 2 y = −2 x + 2

    46.

    y = −3 x + 1 y = −3 x + 1

    47.

    y = x + 2 y = x + 2

    48.

    y = x 3 y = x 3

    49.

    y = x 3 y = x 3

    50.

    y = x 2 y = x 2

    51.

    y = 2 x y = 2 x

    52.

    y = 3 x y = 3 x

    53.

    y = −4 x y = −4 x

    54.

    y = −2 x y = −2 x

    55.

    y = 1 2 x + 2 y = 1 2 x + 2

    56.

    y = 1 3 x 1 y = 1 3 x 1

    57.

    y = 4 3 x 5 y = 4 3 x 5

    58.

    y = 3 2 x 3 y = 3 2 x 3

    59.

    y = 2 5 x + 1 y = 2 5 x + 1

    60.

    y = 4 5 x 1 y = 4 5 x 1

    61.

    y = 3 2 x + 2 y = 3 2 x + 2

    62.

    y = 5 3 x + 4 y = 5 3 x + 4

    63.

    x + y = 6 x + y = 6

    64.

    x + y = 4 x + y = 4

    65.

    x + y = −3 x + y = −3

    66.

    x + y = −2 x + y = −2

    67.

    x y = 2 x y = 2

    68.

    x y = 1 x y = 1

    69.

    x y = −1 x y = −1

    70.

    x y = −3 x y = −3

    71.

    x + y = 4 x + y = 4

    72.

    x + y = 3 x + y = 3

    73.

    x y = 5 x y = 5

    74.

    x y = 1 x y = 1

    75.

    3 x + y = 7 3 x + y = 7

    76.

    5 x + y = 6 5 x + y = 6

    77.

    2 x + y = −3 2 x + y = −3

    78.

    4 x + y = −5 4 x + y = −5

    79.

    2 x + 3 y = 12 2 x + 3 y = 12

    80.

    3 x 4 y = 12 3 x 4 y = 12

    81.

    1 3 x + y = 2 1 3 x + y = 2

    82.

    1 2 x + y = 3 1 2 x + y = 3

    Graph Vertical and Horizontal lines

    In the following exercises, graph the vertical and horizontal lines.

    83.

    x = 4 x = 4

    84.

    x = 3 x = 3

    85.

    x = −2 x = −2

    86.

    x = −5 x = −5

    87.

    y = 3 y = 3

    88.

    y = 1 y = 1

    89.

    y = −5 y = −5

    90.

    y = −2 y = −2

    91.

    x = 7 3 x = 7 3

    92.

    x = 5 4 x = 5 4

    In the following exercises, graph each pair of equations in the same rectangular coordinate system.

    93.

    y=12xy=12x and y=12y=12

    94.

    y=13xy=13x and y=13y=13

    95.

    y=2xy=2x and y=2y=2

    96.

    y=5xy=5x and y=5y=5

    Mixed Practice

    In the following exercises, graph each equation.

    97.

    y=4xy=4x

    98.

    y=2xy=2x

    99.

    y=12x+3y=12x+3

    100.

    y=14x2y=14x2

    101.

    y=xy=x

    102.

    y=xy=x

    103.

    xy=3xy=3

    104.

    x+y=5x+y=5

    105.

    4x+y=24x+y=2

    106.

    2x+y=62x+y=6

    107.

    y=−1y=−1

    108.

    y=5y=5

    109.

    2x+6y=122x+6y=12

    110.

    5x+2y=105x+2y=10

    111.

    x=3x=3

    112.

    x=−4x=−4

    Everyday Math

    113.

    Motor home cost The Robinsons rented a motor home for one week to go on vacation. It cost them $594$594 plus $0.32$0.32 per mile to rent the motor home, so the linear equation y=594+0.32xy=594+0.32x gives the cost, y,y, for driving xx miles. Calculate the rental cost for driving 400,800,and1,200400,800,and1,200 miles, and then graph the line.

    114.

    Weekly earning At the art gallery where he works, Salvador gets paid $200$200 per week plus 15%15% of the sales he makes, so the equation y=200+0.15xy=200+0.15x gives the amount yy he earns for selling xx dollars of artwork. Calculate the amount Salvador earns for selling $900, $1,600,and$2,000,$900, $1,600,and$2,000, and then graph the line.

    Writing Exercises

    115.

    Explain how you would choose three x-valuesx-values to make a table to graph the line y=15x2.y=15x2.

    116.

    What is the difference between the equations of a vertical and a horizontal line?

    Self Check

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

    11.3: Graphing Linear Equations (34)

    After reviewing this checklist, what will you do to become confident for all objectives?

    11.3: Graphing Linear Equations (2024)
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