11_1 - Chapter 11 Systems of Equations and Inequalities...

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Chapter 11 1328 Systems of Equations and Inequalities 11.1 Systems of Linear Equations: Substitution and Elimination 1. 3 x + 4 = 8 - x 4 x = 4 x = 1 The solution set is 1 { } . 2. (a) 3 x + 4 y = 12 x -intercept: 3 x + 4 0 ( 29 = 12 3 x = 12 x = 4 y- intercept: 3 0 ( 29 + 4 y = 12 4 y = 12 y = 3 (b) 3 x + 4 y = 12 4 y = - 3 x + 12 y = - 3 4 x + 3 A parallel line would have slope - 3 4 . 3. inconsistent 4. consistent 5. False 6. True 7. Substituting the values of the variables: 2 x - y = 5 2(2) - ( - 1) = 4 + 1 = 5 5 x + 2 y = 8 5(2) + 2( - = 10 - 2 = 8 Each equation is satisfied, so x = 2, y = - 1 is a solution of the system of equations. 8. Substituting the values of the variables: 3 x + 2 y = 2 3( - 2) + 2(4) = - 6 + 8 = 2 x - 7 y = - 30 ( - 2) - 7(4) = - 2 - 28 = - 30 Each equation is satisfied, so x = - 2, y = 4 is a solution of the system of equations.
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Section 11.1 Systems of Linear Equations: Substitution and Elimination 1329 9. Substituting the values of the variables: 3 x - 4 y = 4 3(2) - 4 1 2 = 6 - 2 = 4 1 2 x - 3 y = - 1 2 1 2 (2) - 3 1 2 = 1 - 3 2 =- 1 2 Each equation is satisfied, so x = 2, y = 1 2 is a solution of the system of equations. 10. Substituting the values of the variables: 2 x + 1 2 y = 0 2 - 1 2 + 1 2 2 ( 29 = - 1 + 1 = 0 3 x - 4 y = - 19 2 3 - 1 2 - 4 2 ( 29 = - 3 2 - 8 = - 19 2 Each equation is satisfied, so x 1 2 , y = 2 is a solution of the system of equations. 11. Substituting the values of the variables: x - y = 3 4 - 1 = 3 1 2 x + y = 3 1 2 (4) + 1 = 2 + 1 = 3 Each equation is satisfied, so x = 4, y = 1 is a solution of the system of equations. 12. Substituting the values of the variables: x - y = 3 - 2 - - 5 ( 29 = - 2 + 5 = 3 - 3 x + y = 1 ( - 3) - 2 ( 29 + - 5 = 6 - 5 = 1 Each equation is satisfied, so x 2, y = - 5 is a solution of the system of equations. 13. Substituting the values of the variables: 3 x + 3 y + 2 z = 4 31 (29 + 3 - 1 ( 29 + 22 ( 29 = 3 - 3 + 4 = 4 x - y - z = 0 1 - - 1 ( 29 - 2 = 1 + 1 - 2 = 0 2 y - 3 z = - 8 2 - 1 ( 29 - 3 2 ( 29 = - 2 - 6 = - 8 Each equation is satisfied, so x = 1, y = - 1, z = 2 is a solution of the system of equations. 14. Substituting the values of the variables: 4 x - z = 7 4 2 ( 29 - 1 = 8 - 1 = 7 8 x + 5 y - z = 0 8 2 ( 29 + 5 - 3 ( 29 - 1 = 16 - 15 - 1 = 0 - x - y + 5 z = 6 ⇒ - 2 - - 3 ( 29 + 5 1 ( 29 = - 2 + 3 + 5 = 6 Each equation is satisfied, so x = 2, y = - 3, z = 1 is a solution of the system of equations.
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Chapter 11 Systems of Equations and Inequalities 1330 15. Substituting the values of the variables: 3 x + 3 y + 2 z = 4 3 2 ( 29 + 3 - 2 ( 29 + 2 2 ( 29 = 6 - 6 + 4 = 4 x - 3 y + z = 10 2 - 3 - 2 ( 29 + 2 = 2 + 6 + 2 = 10 5 x - 2 y - 3 z = 8 5 2 ( 29 - 2 - 2 ( 29 - 3 2 ( 29 = 10 + 4 - 6 = 8 Each equation is satisfied, so x = 2, y = - 2, z = 2 is a solution of the system of equations. 16. Substituting the values of the variables: 4 x - 5 z = 6 4 4 ( 29 - 5 2 ( 29 = 16 - 10 = 6 5 y - z = - 17 5 - 3 ( 29 - 2 ( 29 = - 15 - 2 = - 17 - x - 6 y + 5 z = 24 ⇒ - 4 ( 29 - 6 - 3 ( 29 + 5 2 ( 29 = - 4 + 18 + 10 = 24 Each equation is satisfied, so x = 4, y = - 3, z = 2 is a solution of the system of equations.
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11_1 - Chapter 11 Systems of Equations and Inequalities...

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