913 9.3 Determinant Solution of Linear Systems Cramer’s Rule for Two Equations in Two Variables Given the system a1 x + b1 y = c1 a2 x + b2 y = c2, if D≠0, then the system has the unique solution x = Dx D and y = Dy D , where D= 2 a1 a2 b1 b2 2 , Dx = 2 c1 c2 b1 b2 2 , and Dy = 2 a1 a2 c1 c2 2 . EXAMPLE 5 Applying Cramer’s Rule to a 2 : 2 System Use Cramer’s rule to solve the system of equations. 5x + 7y = -1 6x + 8y = 1 SOLUTION First find D. If D≠0, then find Dx and Dy. D= 2 5 6 7 82 = 5182 - 6172 = -2 Dx = 2 -1 1 7 82 = -1182 - 1172 = -15 Dy = 2 5 6 -1 12 = 5112 - 61-12 = 11 x = Dx D = -15 -2 = 15 2 and y = Dy D = 11 -2 = - 11 2 Cramer’s rule The solution set is E A15 2 , - 11 2 B F . Verify by substituting in the given system. S Now Try Exercise 65. Because graphing calculators can evaluate determinants, they can also be used to apply Cramer’s rule to solve a system of linear equations. The screens above support the result in Example 5. General Form of Cramer’s Rule Let an n * n system have linear equations of the following form. a1x1 + a2 x2 + a3 x3 + g+ an xn = b Define D as the determinant of the n * n matrix of all coefficients of the variables. Define Dx 1 as the determinant obtained from D by replacing the entries in column 1 of D with the constants of the system. Define Dx i as the determinant obtained from D by replacing the entries in column i with the constants of the system. If D≠0, then the unique solution of the system is x1 = Dx 1 D , x2 = Dx 2 D , x3 = Dx 3 D , N, xn = Dx n D . CAUTION Evaluate D first. If D=0, then Cramer’s rule does not apply. The system is inconsistent or has infinitely many solutions.
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