780 CHAPTER 11 Systems of Equations and Inequalities 11.2 Assess Your Understanding 5. True or False The matrix − ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ 1 0 0 3 1 0 2 5 0 is in row echelon form. 6. Multiple Choice Which statement describes the system of equations represented by 1 5 2 0 1 3 0 0 0 3 2 5 ? − − ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ (a) The system has one solution. (b) The system has infinitely many solutions. (c) The system has no solution. (d) The number of solutions cannot be determined. 1. An m by n rectangular array of numbers is called a(n) . 2. The matrix used to represent a system of linear equations is called a(n) matrix. 3. The notation a35 refers to the entry in the row and column of a matrix. 4. Multiple Choice Which matrix is in reduced row echelon form? (a) − − ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ 1 3 2 1 9 1 (b) ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ 1 0 0 1 1 4 (c) ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ 1 0 2 0 9 28 (d) ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ 1 0 2 1 9 4 Concepts and Vocabulary Skill Building In Problems 7–18, write the augmented matrix of the given system of equations. 7. − = + = ⎧ ⎨ ⎪⎪ ⎩ ⎪⎪ x y x y 5 5 4 3 6 8. x y x y 3 4 7 4 2 5 + = − = ⎧ ⎨ ⎪⎪ ⎩ ⎪⎪ 9. x y x y 2 3 6 0 4 6 2 0 + − = − + = ⎧ ⎨ ⎪⎪ ⎩ ⎪⎪ 10. x y x y 9 0 3 4 0 − = − − = ⎧ ⎨ ⎪⎪ ⎩ ⎪⎪ 11. − = + = ⎧ ⎨ ⎪⎪ ⎩ ⎪⎪ x y x y 0.01 0.03 0.06 0.13 0.10 0.20 12. − = − + = ⎧ ⎨ ⎪⎪ ⎪⎪⎪ ⎩ ⎪⎪ ⎪⎪ ⎪ x y x y 4 3 3 2 3 4 1 4 1 3 2 3 13. − + = + = + + = ⎧ ⎨ ⎪⎪ ⎪⎪ ⎩ ⎪⎪ ⎪⎪ x y z x y x y z 10 3 3 5 2 2 14. − − = + = − = ⎧ ⎨ ⎪⎪ ⎪⎪ ⎩ ⎪⎪ ⎪⎪ x y z x y x z 5 0 5 2 3 2 15. + − = − = + − = ⎧ ⎨ ⎪⎪ ⎪⎪ ⎩ ⎪⎪ ⎪⎪ x y z x y x y z 2 3 2 2 5 3 1 16. + − = − + = + − =− ⎧ ⎨ ⎪⎪ ⎪⎪ ⎩ ⎪⎪ ⎪⎪ x y z x z x y z 2 3 4 0 5 2 0 2 3 2 17. x y z x y z x y x y z 10 2 2 1 3 4 5 4 5 0 − − = + + =− − + = − + = ⎧ ⎨ ⎪⎪ ⎪⎪ ⎪⎪ ⎩ ⎪⎪ ⎪⎪ ⎪⎪ 18. − + − = + − + = − − − =− ⎧ ⎨ ⎪⎪ ⎪⎪ ⎩ ⎪⎪ ⎪⎪ x y z w x y z w x y z w 2 5 3 4 2 2 3 5 1 In Problems 19–26, write the system of equations corresponding to each augmented matrix. Then perform the indicated row operation(s) on the given augmented matrix. 19. R r r 1 3 2 5 2 5 2 2 1 2 − − − ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ = − + 20. R r r 1 3 2 5 3 4 2 2 1 2 − − − − ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ = − + 21. R r r R r r 1 3 4 3 5 6 5 3 4 3 6 6 3 5 2 1 2 3 1 3 − − − ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ = − + = + 22. R r r R r r 1 3 3 4 5 3 3 2 4 5 5 6 4 3 2 1 2 3 1 3 − − − − − − − − ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ = + = + 23. R r r R r r 1 3 2 2 5 3 3 6 4 6 4 6 2 3 2 1 2 3 1 3 − − − − − − ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ = − + = + 24. R r r R r r 1 3 4 6 5 6 1 1 4 6 6 6 6 2 1 2 3 1 3 − − − − − − ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ = − + = + 25. R r r R r r 5 3 1 2 5 6 4 1 4 2 2 6 2 2 1 2 1 3 2 3 − − − − − ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ = − + = + 26. R r r R r r 4 3 1 3 5 2 3 6 4 2 6 6 1 2 1 3 2 3 − − − − − ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ = − + = + In Problems 27–38, the reduced row echelon form of a system of linear equations is given. Write the system of equations corresponding to the given matrix. Use x, y; or x, y, z; orx ,x ,x ,x 1 2 3 4 as variables. Determine whether the system is consistent or inconsistent. If it is consistent, give the solution. 27. − ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ 1 0 0 1 5 1 28. − ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ 1 0 0 1 4 0 29. ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ 1 0 0 0 1 0 0 0 0 1 2 3 1. Now Work 1. Modeling 1.ExplainingConcepts Calculus Preview 1.InteractiveFigure
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