SECTION 11.1 Systems of Linear Equations: Substitution and Elimination 763 Finally, back-substitute = − y 2 and = z 1 in equation (1) and solve for x. x y z x x x 1 2 1 1 3 1 2 ( ) + − = − + − − = − − = − = (1) Substitute y 2 =− and z 1. = Simplify. Solve for x. The solution of the original system is = = − = x y z 2, 2, 1 or, using an ordered triplet, ( ) − 2, 2, 1 . You should check this solution. Look back over the solution to Example 9. Note the pattern of eliminating one of the variables from two of the equations, followed by solving the resulting system of two equations and two variables. Although the variables to eliminate is your choice, the method is the same for all systems. Now Work PROBLEM 45 6 Identify Inconsistent Systems of Equations Containing Three Variables 7 Express the Solution of a System of Dependent Equations Containing Three Variables Identifying an Inconsistent System of Linear Equations Solve: + − =− + − =− − + = ⎧ ⎨ ⎪⎪ ⎪⎪ ⎩ ⎪⎪ ⎪⎪ x y z x y z x y z 2 2 2 9 4 1 (1) (2) (3) EXAMPLE 10 Solution Our strategy is the same as in Example 9. However, in this system, it seems easiest to eliminate the variable z first. Do you see why? Multiply equation (1) by −1, and add the result to equation (2). Also, add equations (2) and (3). Now concentrate on the new equations (2) and (3), treating them as a system of two equations containing two variables. Multiply equation (2) by 2, and add the result to equation (3). + − =− + − =− x y z x y z 2 2 2 9 (1) Multiply by 1. − (2) − − + = + − =− − + =− x y z x y z x y 2 2 2 9 7 (1) (2) Add. + − =− − + = − =− x y z x y z x y 2 9 4 1 2 2 8 (2) (3) Add. + − =− − + =− − =− ⎧ ⎨ ⎪⎪ ⎪⎪ ⎩ ⎪⎪ ⎪⎪ x y z x y x y 2 2 7 2 2 8 (1) (2) (3) − + =− − =− x y x y 7 2 2 8 (2) Multiply by 2. (3) − + =− − = − =− x y x y 2 2 14 2 2 8 0 22 (2) (3) Add. + − = − − + = − =− ⎧ ⎨ ⎪⎪ ⎪⎪ ⎩ ⎪⎪ ⎪⎪ x y z x y 2 2 7 0 22 (1) (2) (3) Equation (3) has no solution, so the system is inconsistent. Solving a System of Dependent Equations Solve: − − = − + = − + = ⎧ ⎨ ⎪⎪ ⎪⎪ ⎩ ⎪⎪ ⎪⎪ x y z x y z x y z 2 8 2 3 23 4 5 5 53 (1) (2) (3) EXAMPLE 11 (continued)
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