Section 3.2 Linear Equations in Two Variables 205 The Graph of an Equation in Two Variables The graph of an equation in two variables is the graph of all ordered pair solutions to the equation. The word linear means “relating to or resembling a line.” It is not surprising then that the solution set for any linear equation in two variables forms a line in a rectangular coordinate system. Because two points determine a line, to graph a linear equation it is sufficient to find two solution points and draw the line between them. We will find three solution points and use the third point as a check point. This process is demonstrated in Example 2. Graphing a Linear Equation Graph the equation x 2y 8. Solution: We will find three ordered pairs that are solutions to x 2y 8. To find the ordered pairs, choose an arbitrary value of x or of y.Three choices are recorded in the table. To complete the table, individually substitute each choice into the equation and solve for the missing variable. The substituted value and the solution to the equation form an ordered pair. Example 2 10, 42. Answer 4. 5 4 3 2 1 y 543 2 1 1 2 3 4 5 1 2 3 4 5 x 12, 2 1 , 12 10, 2 x y 2 1 0 From the first row, substitute x 2: x 2y 8 122 2y 8 2y 6 y 3 TIP: Usually we try to choose arbitrary values that will be convenient to graph. From the second row, substitute y 1: x 2y 8 x 2112 8 x 2 8 x 6 From the third row, substitute x 0: x 2y 8 102 2y 8 2y 8 y 4 The completed table is shown with the corresponding ordered pairs. 12, 32 16, 12 10, 42 x y 2 3 6 1 0 4 To graph the equation, plot the three solutions and draw the line through the points (Figure 3-6). Skill Practice 4. Graph the equation 2x y 6. 3 2 21 1 1 2 3 4 5 6 7 1 2 3 4 5 6 7 y x (0, 4) (2, 3) 8 (6, 1) Figure 3-6 Avoiding Mistakes Only two points are needed to graph a line. However, in Example 2, we found a third ordered pair, Notice that this point “lines up” with the other two points. If the three points do not line up, then we know that a mistake was made in solving for at least one of the ordered pairs.
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