Step 2: Determine which variable to eliminate from equations (1) and (3). 12x 18y 9 (1) 4x 6y 3 (3) Eliminate x. Multiply equation (3) by 3. 12x 18y 9 (1) 12x 18y 9 3 times (3) Step 3: Add the equations. 12x 18y 9 12x 18y 9 0 0 True The variables drop out, and we get a true statement. The equations are dependent, so there are an infi nite number of solutions. The solution set is e (x, y) ` y 2 3 x 1 2 f . Notice that, just as in Section 4.2, if the variables drop out and you end up with a true statement, there are an infinite number of solutions. YOU TRY 5 Solve the system using the elimination method. 6x 8y 4 3x 4y 2 3 Use the Elimination Method Twice to Solve a Linear System Sometimes, applying the elimination method twice is the best strategy. EXAMPLE 6 In-Class Example 6 Solve using the elimination method. 5x 3y 6 2x 7y 1 Answer: a 39 41 , 17 41 b Solve using the elimination method. 5x 6y 2 (1) 9x 4y 3 (2) Solution Each equation is written in the form Ax By C, so we begin with Step 2. Step 2: We will eliminate y from equations (1) and (2). Rewrite the System 2(5x 6y) 2(2) 10x 12y 4 3(9x 4 y) 3(3) 27x 12y 9 Step 3: Add the resulting equations to eliminate y. Solve for x. 10x 12y 4 27x 12y 9 37x 5 x 5 37 Solve for x. Normally, we would substitute x 5 37 into equation (1) or equation (2) and solve for y. www.mhhe.com/messersmith SECTION 4.3 Solving Systems by the Elimination Method 267
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