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3.2 Graphing Linear Equations in Two Variables Objective 1 Graph linear equations by plotting ordered pairs. 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 + 2y = 7 are graphed below. 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 + 2y = 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 + 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. 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. Slide 3.2-6 CLASSROOM EXAMPLE 1 Graph Graphing a Linear Equation 5x 2 y 10. Solution: 5 0 2 y 10 2 y 1 0 2 2 y 5 5x 2 0 10 0, 5 5 x 1 0 5 5 x 2 2, 0 5 4 2 y 10 20 2 y 20 10 20 2 y 10 2 2 y5 4, 5 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 3.2-7 CLASSROOM EXAMPLE 2 Graph Graphing a Linear Equation 2 y x 2. 3 Solution: 2 y 0 2 3 y 2 0, 2 2 02 x22 3 2 3 3 2 x 3 2 2 x3 2 4 2 x 2 2 3 2 3 2 x 3 2 3, 0 x 3 3, 4 Slide 3.2-8 Objective 2 Find intercepts. 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 y-intercept 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 x-intercept. To find the y-intercept, let x = 0 and solve for y. Then (0, y) is the yintercept. Slide 3.2-10 CLASSROOM EXAMPLE 3 Finding Intercepts Find the intercepts for the graph of 5x + 2y = 10. Then draw the graph. Solution: 5x 2 0 10 5 x 10 5 5 x2 x-intercept: 2, 0 5 0 2 y 10 2 y 10 2 2 y5 y-intercept: 0,5 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. Slide 3.2-11 Objective 3 Graph linear equations of the form Ax + By = 0. 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 Ax By 0 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. Slide 3.2-13 CLASSROOM EXAMPLE 4 Graphing an Equation with x- and y-Intercepts (0, 0) Graph 4x − 2y = 0. Solution: 12 1 4 6 2 y 0 24 2 y 24 0 24 2 y 24 2 2 y 12 4x 2 2 0 4x 4 4 0 4 4 x 4 4 4 x 1 Slide 3.2-14 Objective 4 Graph linear equations of the form y = k or x = k. 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). Slide 3.2-16 CLASSROOM 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. Slide 3.2-17 CLASSROOM 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. Slide 3.2-18 Graphing linear equations of the form y = k or x = k. (cont’d) Slide 3.2-19 Graphing linear equations of the form y = k or x = k. (cont’d) Slide 3.2-20 Objective 5 Use a linear equation to model data. Slide 3.2-21 CLASSROOM 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. y 32.0 x 684 Solution: 2005 About 850 billion dollars using the graph to estimate. y 32.0 5 684 y 160 684 y 844 Exactly 844 billion dollars using the equation. Slide 3.2-22