Chapter 3 Section 2 3 2 Graphing Linear

Chapter 3 Section 2

3. 2 Graphing Linear Equations in Two Variables Objectives 1 Graph linear equations by plotting ordered pairs. 2 Find intercepts. 3 Graph linear equations of the form Ax + By = 0. 4 Graph linear equations of the form y = k or x = k. 5 Use a linear equation to model data. Copyright © 2012, 2008, 2004 Pearson Education, Inc.

Objective 1 Graph linear equations by plotting ordered pairs. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Slide 3. 2 -3

Graph linear equations by plotting ordered pairs. Infinitely many ordered pairs satisfy a linear equation in two variables. We find these ordered-pair solutions by choosing as many values of x (or y) as we wish and then completing each ordered pair. Some solutions of the equation x + 2 y = 7 are graphed below. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Slide 3. 2 -4

Graph linear equations by plotting ordered pairs. (cont’d) Notice that the points plotted in the previous graph all appear to lie on a straight line, as shown below. Every point on the line represents a solution of the equation x + 2 y = 7, and every solution of the equation corresponds to a point on the line. The line gives a “picture” of all the solutions of the equation x + 2 y = 7. Only a portion of the line is shown, but it extends indefinitely in both directions, suggested by the arrowheads. The line is called the graph of the equation, and the process of plotting the ordered pairs and drawing the line through the corresponding points is called graphing. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Slide 3. 2 -5

Graph linear equations by plotting ordered pairs. (cont’d) Graph of a Linear Equation The graph of any linear equation in two variables is a straight line. Notice the word line appears in the name “linear equation. ” Since two distinct points determine a line, we can graph a straight line by finding any two different points on the line. However, it is a good idea to plot a third point as a check. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Slide 3. 2 -6

EXAMPLE 1 Graphing a Linear Equation Graph Solution: When graphing a linear equation, all three points should lie on the same straight line. If they don’t, double-check the ordered pairs you found. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Slide 3. 2 -7

EXAMPLE 2 Graphing a Linear Equation Graph Solution: Copyright © 2012, 2008, 2004 Pearson Education, Inc. Slide 3. 2 -8

Objective 2 Find intercepts. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Slide 3. 2 -9

Find intercepts. In the previous example, the graph intersects (crosses) the y-axis at (0, − 2) and the x-axis at (3, 0). For this reason (0, − 2) is called the yintercept and (3, 0) is called the x-intercept of the graph. The intercepts are particularly useful for graphing linear equations. They are found by replacing, in turn, each variable with 0 in the equation and solving for the value of the other variable. Finding Intercepts To find the x-intercept, let y = 0 and solve for x. Then (x, 0) is the xintercept. To find the y-intercept, let x = 0 and solve for y. Then (0, y) is the yintercept. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Slide 3. 2 -10

EXAMPLE 3 Finding Intercepts Find the intercepts for the graph of 5 x + 2 y = 10. Then draw the graph. Solution: x-intercept: y-intercept: When choosing x- or y-values to find ordered pairs to plot, be careful to choose so that the resulting points are not too close together. This may result in an inaccurate line. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Slide 3. 2 -11

Objective 3 Graph linear equations of the form Ax + By = 0. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Slide 3. 2 -12

Graph linear equations of the form Ax + By = 0. Line through the Origin If A and B are nonzero real numbers, the graph of a linear equation of the form passes through the origin (0, 0). A second point for a linear equation that passes through the origin can be found as follows: 1. Find a multiple of the coefficients of x and y. 2. Substitute this multiple for x. 3. Solve for y. 4. Use these results as a second ordered pair. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Slide 3. 2 -13

EXAMPLE 4 Graphing an Equation with x- and y-Intercepts (0, 0) Graph 4 x − 2 y = 0. Solution: Copyright © 2012, 2008, 2004 Pearson Education, Inc. Slide 3. 2 -14

Objective 4 Graph linear equations of the form y = k or x = k. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Slide 3. 2 -15

Graphing linear equations of the form y = k or x = k. The equation y = − 4 is the linear equation in which the coefficient of x is 0. Also, x = 3 is a linear equation in which the coefficient of y is 0. These equations lead to horizontal straight lines and vertical straight lines, respectively. Horizontal Line The graph of the linear equation y = k, where k is a real number, is the horizontal line with y-intercept (0, k). There is no y-intercept (unless the vertical line is the y-axis itself). Vertical Line The graph of the linear equation x = k, where k is a real number, is the vertical line with x-intercept (k, 0). There is no x-intercept (unless the horizontal line is the x-axis itself). The equations of horizontal and vertical lines are often confused with each other. Remember that the graph of y = k is parallel to the x-axis and that of x = k is parallel to the y-axis (for k ≠ 0). Copyright © 2012, 2008, 2004 Pearson Education, Inc. Slide 3. 2 -16

EXAMPLE 5 Graphing an Equation of the Form y = k (Horizontal Line) Graph y = − 5. Solution: The equation states that every value of y = − 5. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Slide 3. 2 -17

EXAMPLE 6 Graphing an Equation of the Form x = k (Vertical Line) Graph x − 2 = 0. Solution: After 2 is added to each side the equation states that every value of x = 2. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Slide 3. 2 -18

Graphing linear equations of the form y = k or x = k. (cont’d) Copyright © 2012, 2008, 2004 Pearson Education, Inc. Slide 3. 2 -19

Graphing linear equations of the form y = k or x = k. (cont’d) Copyright © 2012, 2008, 2004 Pearson Education, Inc. Slide 3. 2 -20

Objective 5 Use a linear equation to model data. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Slide 3. 2 -21

Example 7 Using a Linear Equation to Model Credit Card Debt Use (a) the graph and (b) the equation from Example 7 to approximate credit card debt in 2005. Solution: 2005 About 850 billion dollars using the graph to estimate. Exactly 844 billion dollars using the equation. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Slide 3. 2 -22