912 CHAPTER 9 Systems and Matrices Cramer’s Rule The elimination method can be used to develop a process for solving a linear system in two unknowns using determinants. Consider the following system. a1 x + b1 y = c1 (1) a2 x + b2 y = c2 (2) The variable y in this system of equations can be eliminated using multiplication to create coefficients that are additive inverses and adding the two equations. a1b2 x + b1b2 y = c1b2 Multiply (1) by b2. -a2b1 x - b1b2 y = -c2b1 Multiply (2) by -b1. 1a1b2 - a2b12x = c1b2 - c2b1 Add. x = c1b2 - c2b1 a1b2 - a2b1 , if a1b2 - a2b1 ≠0 Similarly, the variable x can be eliminated. -a1a2 x - a2b1 y = -a2c1 Multiply (1) by -a2. a1a2 x + a1b2 y = a1c2 Multiply (2) by a1. 1a1b2 - a2b12y = a1c2 - a2c1 Add. y = a1c2 - a2c1 a1b2 - a2b1 , if a1b2 - a2b1 ≠0 Both numerators and the common denominator of these values for x and y can be written as determinants. c1b2 - c2b1 = 2 c1 c2 b1 b2 2 , a1c2 - a2c1 = 2 a1 a2 c1 c2 2 , and a1b2 - a2b1 = 2 a1 a2 b1 b2 2 The solutions for x and y can be written using these determinants. x = 2 c1 c2 b1 b2 2 2 a1 a2 b1 b2 2 and y = 2 a1 a2 c1 c2 2 2 a1 a2 b1 b2 2 , if 2 a1 a2 b1 b2 2 ≠0. We denote the three determinants in the solution as follows. 2 a1 a2 b1 b2 2 = D, 2 c1 c2 b1 b2 2 = Dx, and 2 a1 a2 c1 c2 2 = Dy These results are summarized as Cramer’s rule. NOTE The elements of D are the four coefficients of the variables in the given system. The elements of Dx are obtained by replacing the coefficients of x in D by the respective constants, and the elements of Dy are obtained by replacing the coefficients of y in D by the respective constants.
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