Wednesday, August 7, 2019
Tax Returns Essay Example for Free
Tax Returns Essay Consider an accountant who prepares tax returns. Suppose a form 1040EZ requires $12 in computer resources to process and 22 minutes of the accountant’s time. Assume a form 1040A takes $25 in computer resources and needs 48 minutes of the accountant’s time. If the accountant can spend $630 on computer resources and has 1194 minutes available, how many forms of 1040EZ and 1040A can the accountant process? Solution: Let      x be the number of form 1040EZ that the accountant can process y be the number of form 1040A that the accountant can process The system of equation required for this problem is 12x + 25y ≠¤ 630 22x + 48y ≠¤ 1194 In augmented matrix form: The reduced form gives us the following set of equations:            x + 2. 5y ≠¤ 52.5          x + 2.5(4.3) ≠¤ 52.5                x ≠¤ 41.7                      y ≠¤ 4.3 Answer: The accountant can process at most 41.7 ( or 41) 1040EZ forms and at most 4.3 (or 4) 1040A forms. You are given the following system of linear equations: x – y + 2z = 13 2x + y – z = -6 -x + 3y + z = -7 a. Provide a coefficient matrix corresponding to the system of linear equations. What is the inverse of this matrix? What is the transpose of this matrix? d. Find the determinant for this matrix. det(A) = (1)(1)(1) + (-1)(-1)(-1) + (2)(2)(3) – (2)(1)(-1) – (-1)(2)(1) – (1)(-1)(3) det(A) = 19 Calculate the following for a. A * B b. -4A c. AT Solve the following linear system using Gaussian elimination. Show work. 3x + y – z = -5 -4x + y = 6 6x – 2y + 3z = 2 Solution: Backward substitution: 19/7 z = 76/7             z = 4 y – 4/7 z = -2/7          y = 2 x+1/3 y – 1/3 z = -5/3            x = -1 Solve the following linear system for x using Cramer’s rule. Show work. x + 2y – 3z = -22 2x – 6y + 8z = 74 -x – 2y + 4z = 29 Solution: The coefficient matrix corresponding to the given system is and the answer column is det(A) = (1)(-6)(4) + (2)(8)(-1) + (-3)(2)(-2) – (-3)(-6)(-1) – (2)(2)(4) – (1)(8)(-2)            = -10 Plug-in the answer column to x column and get the determinant det(X) = (-22)(-6)(4) + (2)(8)(29) + (-3)(74)(-2) – (-3)(-6)(29) – (2)(74)(4) – (-22)(8)(-2)            = -30 Plug-in the answer column to y column and get the determinant det(Y) = (1)(74)(4) + (-22)(8)(-1) + (-3)(2)(29) – (-3)(74)(-1) – (-22)(2)(4) – (1)(8)(29)            = 20 Plug-in the answer column to z column and get the determinant det(Z) = (1)(-6)(29) + (2)(74)(-1) + (-22)(2)(-2) – (-22)(-6)(-1) – (2)(2)(29) – (1)(74)(-2)            = -70 By Cramer’s rule, the solution to the system is x = -30 / -10 = 3 y = 20 / -10 = -2 z = -70 / -10 = 7
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