354 CHAPTER 6 Algebra, Graphs, and Functions Fig. 6.25 shows the system of equations in Example 2, x y 4 + = and x y 2 1, − = − graphed on a Texas Instrument TI-84 Plus calculator. In Example 2 we were able to estimate the solution to a system of linear equations by determining what appeared to be the point of intersection of the graphs in the systems of linear equations. Since the solution to a system of equations may not be integer values, you may not be able to obtain the exact solution by graphing. If an exact answer to a system of linear equations is needed, we can solve the system with algebraic methods. Two such methods are the substitution method and the addition method. We first discuss the substitution method. SOLVING A SYSTEM OF LINEAR EQUATIONS BY GRAPHING 1. Determine three ordered pairs that satisfy each equation. 2. Plot the points that correspond to the ordered pairs and graph both equations on the same axes. 3. The coordinates of the point or points of intersection of the graphs are the solution or solutions to the system of equations. PROCEDURE y x 5 4 3 2 1 1 2 3 4 5 22 22 23 24 23 24 (–2, –3) 6 6 x 1 y 5 4 2x 2 y 5 21 (0, 4) (0, 1) (4, 0) (1, 3) 21 21 Figure 6.24 x y 2 1 − = − x y 2− 3− 0 1 1 3 The graphs intersect at (1, 3), which is the solution to the system of equations. This point is the only point that satisfies both equations. CHECK: + = + = = x y 4 1 3 4 4 4 True − = − − = − − = − − = − x y 2 1 2(1) 3 1 2 3 1 1 1 True 7 Figure 6.25 Example 2 A Unique Solution by Graphing Determine the solution to the following system of equations graphically. + = − = − x y x y 4 2 1 Solution For each equation, substitute values for x and determine the corresponding values for y. Three ordered pairs that satisfy + = x y 4 are (0, 4), (1, 3), and (4, 0), as shown in the left hand table below. Three ordered pairs that satisfy x y 2 1 − = − are − − ( 2, 3), (0, 1), and (1, 3), as shown in the right-hand table below. Next, graph both x y 4 + = and x y 2 1 − = − on the same axes, Fig. 6.24. x y 4 + = x y 0 4 1 3 4 0 Now try Exercise 15
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