Definition/Procedure Example If the variables drop out and a false equation is obtained, the system has no solution. The system is inconsistent, and the solution set is . (p. 258) Solve by the substitution method. 2x 8y 9 x 4y 2 Step 1: The second equation is solved for x. Step 2: Substitute 4y 2 for x in the fi rst equation. 2(4y 2) 8y 9 Step 3: Solve the equation above for y. 2(4y 2) 8y 9 8y 4 8y 9 Distribute. 4 9 False The system has no solution. The solution set is . If the variables drop out and a true equation is obtained, the system has an infi nite number of solutions. The equations are dependent. (p. 259) Solve by the substitution method. y x 3 3x 3y 9 Step 1: The fi rst equation is solved for y. Step 2: Substitute x 3 for y in the second equation. 3x 3(x 3) 9 Step 3: Solve the equation above for x. 3x 3(x 3) 9 3x 3x 9 9 Distribute. 9 9 True The system has an infi nite number of solutions of the form {(x, y)y x 3}. 4.3 Solving Systems by the Elimination Method Steps for Solving a System of Two Linear Equations by the Elimination Method Step 1: Write each equation in the form Ax By C. Step 2: Determine which variable to eliminate. If necessary, multiply one or both of the equations by a number so that the coeffi cients of the variable to be eliminated are negatives of one another. Step 3: Add the equations, and solve for the remaining variable. Step 4: Substitute the value found in Step 3 into either of the original equations to fi nd the value of the other variable. Step 5: Check the solution in each of the original equations. (p. 264) Solve using the elimination method. 4x 5y 7 5x 6y 8 Eliminate x. Multiply the fi rst equation by 5, and multiply the second equation by 4 to rewrite the system with equivalent equations. Rewrite the system 5(4x 5y) 5(7) S 20x 25y 35 4(5x 6y) 4(8) S 20x 24y 32 Add the equations: 20x 25y 35 20x 24y 32 y 3 Substitute y 3 into either of the original equations, and solve for x. 4x 5y 7 4x 5(3) 7 4x 15 7 4x 8 x 2 The solution is (2, 3). Verify this by substituting (2, 3) into each of the original equations. 300 CHAPTER 4 Linear Equations and Inequalities in Two Variables www.mhhe.com/messersmith
messersmith_power_introductory_algebra_1e_ch4_7_10
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