790 CHAPTER 11 Systems of Equations and Inequalities Notice the similarity between this pattern and the pattern observed earlier for a system of two equations containing two variables. Cramer’s Rule with Inconsistent or Dependent Systems • If = D 0 and at least one of the determinants D D , , x y or Dz is different from 0, then the system is inconsistent and the solution set is Ø, or { } . • If = D 0 and all the determinants D D , , x y and Dz equal 0, then the system is consistent and dependent, so there are infinitely many solutions. The system must be solved using row reduction techniques. Using Cramer’s Rule Use Cramer’s Rule, if applicable, to solve the following system: + − = − + + =− − − = ⎧ ⎨ ⎪⎪ ⎪⎪ ⎩ ⎪⎪ ⎪⎪ (1) (2) (3) x y z x y z x y z 2 3 2 4 3 2 3 4 Solution EXAMPLE 5 ( ) ( ) ( ) ( ) ( ) ( ) = − − − − = − ⋅ ⋅ − − + − ⋅ ⋅ − − + − − − − =⋅−−+−⋅=+= + + + D 2 1 1 1 2 4 1 2 3 1 2 2 4 2 3 1 1 1 4 1 3 1 1 1 2 1 2 2 2 1 1 1 0 4 1 5 1 1 1 2 1 3 Because ≠ D 0, proceed to find the values of D D , , x y and D.z To find D ,x replace the coefficients of x in D with the constants and then evaluate the determinant. ( ) ( ) ( ) ( ) ( ) ( )( ) = − − − − = − ⋅ ⋅ − − + − ⋅ ⋅ − − + − − − − = ⋅ − − + − − = + + + D 3 1 1 3 2 4 4 2 3 1 3 2 4 2 3 1 1 3 4 4 3 1 1 3 2 4 2 3 2 1 7 1 2 15 x 1 1 1 2 1 3 ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) = − − − − = − ⋅ ⋅ − − + − ⋅ ⋅ − − + − − − − = − − − + − − = − + + + D 2 3 1 1 3 4 1 4 3 1 2 3 4 4 3 1 3 1 4 1 3 1 1 1 3 1 4 2 7 3 1 1 1 10 y 1 1 1 2 1 3 ( ) ( ) ( ) ( ) = − − − = − ⋅ ⋅ − − + − ⋅ ⋅ − − + − ⋅ ⋅ − − = ⋅ − − + ⋅ = + + + D 2 1 3 1 2 3 1 2 4 1 2 2 3 2 4 1 1 1 3 1 4 1 3 1 2 1 2 2 2 1 1 3 0 5 z 1 1 1 2 1 3 The value of the determinant D of the coefficients of the variables is As a result, = = = = = − = − = = = x D D y D D z D D 15 5 3 10 5 2 5 5 1 x y z The solution is = = − = x y z 3, 2, 1 or, using an ordered triplet, ( ) − 3, 2, 1 . Cramer’s Rule cannot be used when the determinant of the coefficients of the variables, D, is 0. But can anything be learned about the system other than it is not a consistent and independent system if = D 0? The answer is yes! Now Work PROBLEM 33
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