3.2 Graphing Linear Equations in Two Variables

Objective 1 Graph linear equations by plotting ordered pairs. Slide 3.2-3

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 Graph linear equations by plotting ordered pairs.

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 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. 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 “linear equation.” Slide Graph linear equations by plotting ordered pairs. (cont’d)

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. Graph Slide Graphing a Linear Equation CLASSROOM EXAMPLE 1

Solution: Graph Slide Graphing a Linear Equation CLASSROOM EXAMPLE 2

Objective 2 Find intercepts. Slide 3.2-9

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. Finding 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. They are found by replacing, in turn, each variable with 0 in the equation and solving for the value of the other variable. To find the y-intercept, let x = 0 and solve for y. Then (0, y) is the y- intercept. Slide Find intercepts.

Solution: Find the intercepts for the graph of 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. Slide Finding Intercepts CLASSROOM EXAMPLE 3

Objective 3 Graph linear equations of the form Ax + By = 0. Slide

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. Slide

Solution: Graph 4x − 2y = 0. Slide Graphing an Equation with x- and y-Intercepts (0, 0) CLASSROOM EXAMPLE 4

Objective 4 Graph linear equations of the form y = k or x = k. Slide

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). 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 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).

Solution: The equation states that every value of y = − 5. Graph y = − 5. Slide Graphing an Equation of the Form y = k (Horizontal Line) CLASSROOM EXAMPLE 5

Solution: After 2 is added to each side the equation states that every value of x = 2. Graph x − 2 = 0. Slide Graphing an Equation of the Form x = k (Vertical Line) CLASSROOM EXAMPLE 6

Slide Graphing linear equations of the form y = k or x = k. (cont’d)

Slide Graphing linear equations of the form y = k or x = k. (cont’d)

Objective 5 Use a linear equation to model data. Slide

Solution: About 850 billion dollars using the graph to estimate. Use (a) the graph and (b) the equation from Example 7 to approximate credit card debt in Exactly 844 billion dollars using the equation. Slide Using a Linear Equation to Model Credit Card Debt CLASSROOM EXAMPLE 7