784 CHAPTER 11 Systems of Equations and Inequalities OBJECTIVES 1 Evaluate 2 by 2 Determinants (p. 784) 2 Use Cramer’s Rule to Solve a System of Two Equations Containing Two Variables (p. 785) 3 Evaluate 3 by 3 Determinants (p. 787) 4 Use Cramer’s Rule to Solve a System of Three Equations Containing Three Variables (p. 789) 5 Know Properties of Determinants (p. 791) The previous section described a method of using matrices to solve a system of linear equations. This section describes yet another method for solving systems of linear equations; however, it can be used only when the number of equations equals the number of variables. This method, called Cramer’s Rule , is based on the concept of a determinant . Although the method works for all systems where the number of equations equals the number of variables, it is most often used for systems of two equations containing two variables or three equations containing three variables. 1 Evaluate 2 by 2 Determinants 11.3 Systems of Linear Equations: Determinants = − a b c d ad bc Minus bc ad DEFINITION 2 by 2 Determinant If a, b, c, and d are four real numbers, the symbol = D a b c d is called a 2 by 2 determinant . Its value is the number − ad bc; that is, = = − D a b c d ad bc (1) The following illustration may be helpful for remembering the value of a 2 by 2 determinant: EXAMPLE 1 Evaluating a 2 by 2 Determinant Evaluate: − 3 2 6 1 Algebraic Solution Graphing Solution First, enter the matrix whose entries are those of the determinant into the graphing utility and name it A. Using the determinant command, obtain the result shown in Figure 10 on a TI-84 Plus CE graphing calculator. ( ) ( ) − = ⋅ − − = − − = 3 2 6 1 3 1 6 2 3 12 15 Now Work PROBLEM 7 Figure 10
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