876 CHAPTER 9 Systems and Matrices Special Systems The systems in Examples 1 and 2 had a single solution. The system in the next example has no solution. EXAMPLE 2 Solving a System (Elimination Method) Solve the system. 3x - 4y = 1 (1) 2x + 3y = 12 (2) SOLUTION One way to eliminate a variable is to use the second transformation and multiply each side of equation (2) by -3, to obtain an equivalent system. 3x - 4y = 1 (1) -6x - 9y = -36 Multiply (2) by -3. (3) Now multiply each side of equation (1) by 2, and use the third transformation to add the result to equation (3), eliminating x. Solve the result for y. 6x - 8y = 2 Multiply (1) by 2. -6x - 9y = -36 (3) -17y = -34 Add. y = 2 Solve for y. Substitute 2 for y in either of the original equations and solve for x. 3x - 4y = 1 (1) 3x - 4122 = 1 Let y = 2 in (1). 3x - 8 = 1 Multiply. 3x = 9 Add 8. x = 3 Divide by 3. A check shows that 13, 22 satisfies both equations (1) and (2). Therefore, the solution set is 513, 226. The graph in Figure 2 confirms this. S Now Try Exercise 21. Write the x-value first. 2x + 3y = 12 3x – 4y = 1 (3, 2) 6 3 –1 x 4 0 y Consistent system with solution set {(3, 2)} Figure 2 EXAMPLE 3 Solving an Inconsistent System Solve the system. 3x - 2y = 4 (1) -6x + 4y = 7 (2) SOLUTION To eliminate the variable x, multiply each side of equation (1) by 2, and add the result to equation (2). 6x - 4y = 8 Multiply (1) by 2. -6x + 4y = 7 (2) 0 = 15 False Since 0 = 15 is false, the system is inconsistent and has no solution. As suggested by Figure 3, this means that the graphs of the equations of the system never intersect. (The lines are parallel.) The solution set is ∅. S Now Try Exercise 31. 2 –2 x 2 0 3x – 2y = 4 –6x + 4y = 7 y Inconsistent system with solution set ∅ Figure 3
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