756 CHAPTER 11 Systems of Equations and Inequalities (c) + + = − + = − − = ⎧ ⎨ ⎪⎪ ⎪⎪ ⎩ ⎪⎪ ⎪⎪ x y z x y z x y z 6 3 2 4 9 0 (d) + + = − = ⎧ ⎨ ⎪⎪ ⎩ ⎪⎪ x y z x y 5 2 (e) + + = + = + = = ⎧ ⎨ ⎪⎪ ⎪⎪ ⎪ ⎩ ⎪⎪ ⎪⎪ ⎪ x y z x z y z x 6 2 2 4 2 4 (1) Three equations containing three variables, x y , , and z (2) (3) (1) Two equations containing three variables, x y , , and z (2) (1) Four equations containing three variables, x y , , and z (2) (3) (4) We use a brace to remind us that we are dealing with a system of equations, and we number each equation in the system for convenient reference. A solution of a system of equations consists of values for the variables that are solutions of each equation of the system. To solve a system of equations means to find all solutions of the system. For example, = = x y 2, 1 is a solution of the system in Example 2(a), because + = − + =− ⎧ ⎨ ⎪⎪ ⎩ ⎪⎪ ⋅ + = + = − ⋅ + ⋅ =− + =− ⎧ ⎨ ⎪⎪ ⎩⎪⎪ x y x y 2 5 4 6 2 2 2 1 4 1 5 4 2 6 1 8 6 2 (1) (2) This solution may also be written as the ordered pair ( ) 2, 1 . A solution of the system in Example 2(b) is = = x y 1, 2, because + = + = ⎧ ⎨ ⎪⎪⎪ ⎩ ⎪⎪⎪ + = + = ⋅ + = + = ⎧ ⎨ ⎪⎪ ⎩⎪⎪ x y x y 5 2 4 1 2 1 4 5 2 1 2 2 2 4 2 2 (1) (2) Another solution of the system in Example 2(b) is = = − x y 11 4 , 3 2 , which you can check for yourself. A solution of the system in Example 2(c) is = = = x y z 3, 2, 1, because (1) (2) (3) + + = − + = − − = ⎧ ⎨ ⎪⎪ ⎪⎪ ⎩ ⎪⎪ ⎪⎪ + + = ⋅ − ⋅ + ⋅ = − + = − − = ⎧ ⎨ ⎪⎪ ⎪ ⎩ ⎪⎪ ⎪⎪ x y z x y z x y z 6 3 2 4 9 0 3 2 1 6 332241944 9 3 2 1 0 This solution may also be written as the ordered triplet ( ) 3, 2, 1 . Note that = = = x y z 3, 3, 0 is not a solution of the system in Example 2(c). (1) (2) (3) + + = − + = − − = ⎧ ⎨ ⎪⎪ ⎪⎪ ⎩ ⎪⎪ ⎪⎪ + + = ⋅ − ⋅ + ⋅ = ≠ − − = ⎧ ⎨ ⎪⎪ ⎪ ⎩ ⎪⎪ ⎪⎪ x y z x y z x y z 6 3 2 4 9 0 3 3 0 6 3 3 2 3 4 0 3 9 3 3 0 0 Although = = = x y z 3, 3, and 0 satisfy equations (1) and (3), they do not satisfy equation (2). Any solution of the system must satisfy each equation of the system. Now Work PROBLEM 11 When a system of equations has at least one solution, it is said to be consistent. When a system of equations has no solution, it is called inconsistent. An equation in n variables is said to be linear if it is equivalent to an equation of the form + + + = a x a x a x b n n 1 1 2 2 where … x x x , , , n 1 2 are n distinct variables, … a a a b , , , , n 1 2 are constants, and at least one of the a’s is not 0.
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