914 CHAPTER 9 Systems and Matrices EXAMPLE 6 Applying Cramer’s Rule to a 3 : 3 System Use Cramer’s rule to solve the system of equations. x + y - z + 2 = 0 2x - y + z + 5 = 0 x - 2y + 3z - 4 = 0 SOLUTION x + y - z = -2 2x - y + z = -5 x - 2y + 3z = 4 Verify the required determinants. D= 3 1 2 1 1 -1 -2 -1 1 3 3 = -3, Dx = 3 -2 -5 4 1 -1 -2 -1 1 3 3 = 7, Dy = 3 1 2 1 -2 -5 4 -1 1 3 3 = -22, Dz = 3 1 2 1 1 -1 -2 -2 -5 4 3 = -21 x = Dx D = 7 -3 = - 7 3 , y = Dy D = -22 -3 = 22 3 , z = Dz D = -21 -3 = 7 The solution set is E A - 7 3 , 22 3 , 7B F. S Now Try Exercise 81. Rewrite each equation in the form ax + by + cz + g= k. CAUTION As shown in Example 6, each equation in the system must be written in the form ax + by + cz + g= k before Cramer’s rule is used. EXAMPLE 7 Showing That Cramer’s Rule Does Not Apply Show that Cramer’s rule does not apply to the following system. 2x - 3y + 4z = 10 6x - 9y + 12z = 24 x + 2y - 3z = 5 SOLUTION We need to show that D= 0. Expand about column 1. D= 3 2 6 1 -3 -9 2 4 12 -3 3 = 22 -9 2 12 -32 - 62 -3 2 4 -3 2 + 12 -3 -9 4 122 = 2132 - 6112 + 1102 = 0 Because D= 0, Cramer’s rule does not apply. S Now Try Exercise 77. NOTE When D=0, the system is either inconsistent or has infinitely many solutions. Use the elimination method to tell which is the case. Verify that the system in Example 7 is inconsistent, and thus the solution set is ∅.
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