358 CHAPTER 6 Algebra, Graphs, and Functions Now substitute 3 for x in either of the original equations to determine the value of y. + = + = + = = x y y y y 4 13 4(3) 13 12 13 1 The solution to the system is (3, 1). 7 Now try Exercise 35 Example 7 Multiplying One Equation While Using the Addition Method Solve the following system of equations by the addition method. − = + = x y x y 4 2 3 2 7 Solution We can multiply the top equation by 2 and then add the two equations to eliminate the variable y. − = + = x y x y 2[4 2] 3 2 7 gives − = + = x y x y 8 2 4 3 2 7 − = + = = = x y x y x x 8 2 4 3 2 7 11 11 1 Now we determine y by substituting 1 for x in either of the original equations. − = − = − = − = − = x y y y y y 4 2 4(1) 2 4 2 2 2 The solution to the system is (1, 2). 7 Now Try Exercise 37 Example 8 Multiplying Both Equations While Using the Addition Method Solve the following system of equations by the addition method. x y x y 3 4 8 2 3 9 − = + = Solution In this system, we cannot eliminate a variable by multiplying only one equation by an integer value and then adding. To eliminate a variable, we can multiply each equation by a different number. To eliminate the variable x, we can multiply the top equation by 2 and the bottom equation by 3− (or the top by 2− and the bottom by 3) and then add the two equations. − = − = − + = − − = − x y x y x y x y 2[3 4 8] gives 6 8 16 3[2 3 9] gives 6 9 27 x y x y y y 6 8 16 6 9 27 17 11 11 17 − = − − = − − = − = Timely Tip Following is a summary of the different types of systems of linear equations. • A consistent system of equations is one that has at least one solution. • An inconsistent system of equations is one that has no solution. • A dependent system of equations is one that has an infinite number of solutions.
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