6.7 Solving Systems of Linear Equations 353 Solving a System of Linear Equations by Graphing To solve a system of linear equations graphically, we graph both equations on the same axes. If the system has a unique solution, then the solution will be the ordered pair that represents the point of intersection of the lines in the system. When two lines are graphed, three situations are possible, as illustrated in Fig. 6.23. y x y x y x Solution Consistent Inconsistent Dependent Line 1 Line 1 Intersects Line 2 • Exactly one solution Line 1 Is Parallel to Line 2 • No Solution Line 1 Is the Same Line as Line 2 • Infinite number of solutions • The point of intersection represents the solution. • Since Parallel lines do not intersect, there is no solution. • Every point on the common line represents a solution. • Consistent system • Inconsistent system • Dependent system • Lines have different slopes • Lines have same slopes and different y-intercepts • Lines have same slope and same y-intercept Line 2 Line 1 Line 2 Line 1 Line 2 Figure 6.23 In addition to the information given in Fig. 6.23, a consistent system has at least one solution. Since a dependent system has at least one solution, a dependent system is also a consistent system. Each situation shown in Fig. 6.23 will be illustrated in upcoming examples. We next provide a procedure for solving a system of linear equations by graphing. b) For − (12, 9) we substitute = x 12 and y 9 = − into each equation in the system. = − − = − − = − − = y x 1 2 4 9 1 2 (12) 4 9 6 4 9 2 + = − + − = − − = − − = − x y 3 5 9 3(12) 5( 9) 9 36 45 9 9 9 False True Since − (12, 9) does not satisfy both equations, it is not a solution to the system. c) For − (2, 3) we substitute = x 2 and y 3 = − into each equation in the system. = − − = − − = − − = − y x 1 2 4 3 1 2 (2) 4 3 1 4 3 3 + = − + − = − − = − − = − x y 3 5 9 3(2) 5( 3) 9 6 15 9 9 9 True True Since − (2, 3) satisfies both equations, it is a solution to the system. 7 Now try Exercise 11 Solving a Linear System by Graphing
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