902 CHAPTER 9 Systems and Matrices Each augmented matrix is in row-echelon form. Assume that the variables are x, y, and z and use back-substitution to obtain the solution of the associated system of linear equations. 49. c 1 1 0 1 2 5 3d 50. c 1 3 2 0 1 2 2 1d 51. £ 1 2 2 0 1 1 0 0 1 3 26 11 6§ 52. £ 1 1 4 0 1 2 0 0 1 3 6 4 1 2§ 53. D 1 -1 3 - 1 3 0 1 -5 4 0 0 1 4 8 3 7 4 -3T 54. £ 1 -3 -1 0 1 -1 4 0 0 1 3 -3 1 2 2§ Use the Gaussian elimination method to solve each system of equations. For systems in three variables with infinitely many solutions, write the solution set with z arbitrary. See Example 5. 55. 4x - 4y + 4z = -8 x - 2y - 2z = -1 2x + y + 3z = 1 56. x + 2y + 2z = 11 x - y - z = -4 2x + 5y + 9z = 39 57. x - 3y + 4z = 1 4x - 10y + 10z = 4 -3x + 9y - 5z = -6 58. 2x + y - 6z = 1 4x + 2y + 9z = 9 6x - y + 12z = 5 59. x + 2y + z + w = 2 x - y + z - w = 4 3x + 4y + 2z - w = 8 2x + 3y + 4z + 5w = 5 60. -3x + 4y - 5z + 2w = 0 -2x + 7y - 3z + 3w = 2 x - y + z + w = 1 -3x - 6y + 4z - w = 6 61. x - y + 3z = 3 -2x + 3y - 11z = -4 x - 2y + 8z = 6 62. x + 2y + 3z = 4 5x + 6y + 7z = 8 2x + 4y + 6z = 9 63. x - 3y + z = 5 -2x + 7y - 6z = -9 x - 2y - 3z = 6 64. -3x + 4y - z = -4 x + 2y + z = 4 -12x + 16y - 4z = -16 For each equation, determine the constants A and B that make the equation an identity. (Hint: Combine terms on the right, and set coefficients of corresponding terms in the numerators equal.) 65. 1 1x - 121x + 12 = A x - 1 + B x + 1 66. x + 4 x2 = A x + B x2 67. x 1x - a21x + a2 = A x - a + B x + a 68. 2x 1x + 221x - 12 = A x + 2 + B x - 1
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