204 Chapter 3 Graphing Linear Equations in Two Variables 4 3 2 321 1 1 2 3 Answers 1. No 2. Yes 3. Yes (4, 0) 5 54 4 5 1 2 3 4 (1, 5) (1, 3) (2, 2) 5 y x Figure 3-5 The equation is a linear equation in two variables.A solution to such an equation is an ordered pair (x, y) that makes the equation a true statement. Several solutions to the equation x y 4 are listed here: 1x, y2 x y 4 12, 22 122 122 4 11, 32 112 132 4 14, 02 142 102 4 By graphing these ordered pairs, we see that the solution points line up (Figure 3-5). Notice that there are infinitely many solutions to the equation so they cannot all be listed. Therefore, to visualize all solutions to the equation we draw the line through the points in the graph. Every point on the x y 4, line represents an ordered pair solution to the equation x y 4, and the line represents the set of all solutions to the equation. Determining Solutions to a Linear Equation Example 1 For the linear equation, 6x 5y 12, determine whether the given ordered pair is a solution. a. (2, 0) b. (3, 1) c. Solution: a. 6x 5y 12 a1, 6 5 b 6122 5102 12 Substitute x 2 and y 0. ✔ True The ordered pair (2, 0) is a solution. b. 12 0 12 6x 5y 12 6132 5112 12 x 3 y 1. Substitute and The ordered pair (3, 1) is not a solution. c. Substitute x 1 and y 18 5 12 6x 5y 12 6 5 b 12 6 6 12 b a1, 6 5 ✔ True The ordered pair is a solution. Skill Practice Given the equation 3x 2y 12 , determine whether the given ordered pair is a solution. 1. (4, 0) 2. (2, 3) 3. a1, 15 2 b 2. Graphing Linear Equations in Two Variables by Plotting Points In this section, we will graph linear equations in two variables. 6 5 6112 5 . a x y 4 x y 4 Solution: Check: ✔ ✔ ✔ 11, 52 112 152 4 ✔
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