Chapter 3 Section 2 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Graphing Linear Equations in Two Variables Graph linear equations by plotting ordered pairs. Find intercepts. Graph linear equations of the form Ax + By = 0. Graph linear equations of the form y = k or x = k. Use a linear equation to model data.

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 1 Objective 1 Slide Graph linear equations by plotting ordered pairs.

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 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 + 2y = 7 are graphed below. Slide

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 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 + 2y = 7, and every solution of the equation corresponds to a point on the line. Slide The line gives a “picture” of all the solutions of the equation x + 2y = 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 © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Graph linear equations by plotting ordered pairs. (cont’d) In summary, the graph of any linear equation in two variables is a straight line. Slide 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. Notice the word line appears in the name “lineear equation.”

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 1 Graphing a Linear Equation 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. Slide Graph

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 2 Graphing a Linear Equation Solution: Slide Graph

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 2 Objective 2 Find intercepts. Slide

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 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 y-intercept and (3,0) is called the x-intercept of the graph. Find intercepts. To find the x-intercept, let y = 0 and solve for x. Then (x,0) is the x-intercept. The intercepts are particularly useful for graphing linear equations. The intercepts are found by replacing, in turn, each variable with 0 in the equation and solving for the value of the other variable. Slide To find the y-intercept, let x = 0 and solve for y. Then (0, y) is the y-intercept.

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 3 Solution: Finding Intercepts Slide Find the intercepts for 5x + 2y = 10. Then draw the graph. y-intercept:x-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 © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 3 Objective 3 Graph linear equations of the form Ax + By = 0. Slide

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley If A and B are nonzero real numbers, the graph of a linear equation of the form passes through the origin (0,0). Graph linear equations of the form Ax + By = 0. Slide 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 © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 4 Solution: Graphing an Equation of the Form Ax + By = 0 Slide Graph 4x − 2 = 0.

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 4 Objective 4 Graph linear equations of the form y = k or x = k. Slide

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Graphing linear equations of the form y = k or x = k. Slide 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 and vertical straight lines, respectively. The graph of the linear equation y = k, where k is a real number, is a horizontal line with y-intercept (0, k) and no x-intercept. The graph of the linear equation x = k, where k is a real number, is a vertical line with x-intercept (k,0) and no y-intercept. 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.

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 5 Graphing an Equation of the Form y = k Solution: The equation states that every value of y = −5. Slide Graph y = −5.

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 6 Solution: After 2 is added to each side the equation states that every value of x = 2. Graphing an Equation of the Form x = k Slide Graph x − 2 = 0.

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 5 Objective 5 Use a linear equation to model data. Slide

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley The different forms of linear equations from this section and the methods of graphing them are given in the following summary. Slide

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 7 Solution: a) about 525 billion dollars Use a Linear Equation to Model Credit Card Debt Slide Use a) the graph and b) the equation to approximate credit card debt in 1997, where x = billion dollars. b)