Solve the system by substitution. 2x 6y 10 (1) x 3y 5 (2) Solution Step 1: Equation (2) is already solved for x. Step 2: 2x 6y 10 Substitute 3y 5 for x in equation (1). 2(3y 5) 6y 10 Step 3: 2(3y 5) 6y 10 Solve the resulting equation for y. 6y 10 6y 10 Distribute. 10 10 True Because the variables drop out and we get a true statement, the system has an infi nite number of solutions. The equations are dependent, and the solution set is {(x, y)|x 3y 5}. x y 5 5 5 5 2x 6y 10 x 3y 5 The graph shows that the equations in the system are the same line; therefore, the system has an infi nite number of solutions. Note When you are solving a system of equations and the variables drop out: 1) If you get a false statement, like 20 7, then the system has no solution and is inconsistent. 2) If you get a true statement, like 10 10, then the system has an infinite number of solutions. The equations are dependent. YOU TRY 3 Solve each system by substitution. a) 20x 5y 3 b) x 3y 5 4x y 1 4x 12y 20 ANSWERS TO YOU TRY EXERCISES 1) a 2 3 , 1b 2) a) (8, 2) b) (1, 5) 3) a) b) {(x, y)|x 3y 5} EXAMPLE 5 In-Class Example 5 Solve the system by substitution. x 12 2y 2x 4y 24 Answer: infinite number of solutions of the form {(x, y)0x 2y 12} Remember that a system of linear equations has an infinite number of solutions when the two equations represent the same line. www.mhhe.com/messersmith SECTION 4.2 Solving Systems by the Substitution Method 259
messersmith_power_introductory_algebra_1e_ch4_7_10
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