Math Problem Statement

Pivot the 9 to obtain a new tableau and corresponding new solutions in bracket form

Pivot the 9 to obtain a new tableau and corresponding new solutions in bracket form

Solution

Let's tackle each part of this problem step-by-step, focusing initially on the pivot operation required for part (b).

The tableau provided is as follows:

2 & 9 & 1 & 0 & 0 & 198 \\ 7 & 1 & 0 & 1 & 0 & 749 \\ -10 & -6 & 0 & 0 & 1 & 0 \\ \end{bmatrix}$$ This tableau represents a linear programming problem in the standard simplex tableau form, where the columns correspond to the variables $$x$$, $$y$$, $$u$$, $$v$$, and $$M$$, and the last column represents the constants (right-hand side values). ### Step-by-Step Solution #### Part (b) - Pivot on 9 (element in row 1, column 2) 1. **Identify the pivot element**: The pivot element is $$9$$, located at row 1, column 2. 2. **Row operations** to make the pivot element $$1$$ and other elements in the pivot column $$0$$. - **Row 1 (Make pivot 1)**: Divide row 1 by 9. \[ R_1 = \frac{1}{9} R_1 \Rightarrow \left[\frac{2}{9}, 1, \frac{1}{9}, 0, 0, \frac{198}{9}\right] = \left[\frac{2}{9}, 1, \frac{1}{9}, 0, 0, 22\right] \] - **Row 2 (Eliminate column entry)**: Update row 2 to make the entry in the $$y$$-column (second column) zero. \[ R_2 = R_2 - (1) \times R_1 = \left[7, 1, 0, 1, 0, 749\right] - \left[\frac{2}{9}, 1, \frac{1}{9}, 0, 0, 22\right] \] Simplifying, $$R_2 = \left[\frac{61}{9}, 0, -\frac{1}{9}, 1, 0, 727\right]$$ - **Row 3 (Eliminate column entry)**: Update row 3 similarly to make the entry in the $$y$$-column zero. \[ R_3 = R_3 + 6 \times R_1 = \left[-10, -6, 0, 0, 1, 0\right] + 6 \times \left[\frac{2}{9}, 1, \frac{1}{9}, 0, 0, 22\right] \] Simplifying, \[ R_3 = \left[\frac{-10 + 12}{9}, 0, \frac{6}{9}, 0, 1, 132\right] = \left[\frac{2}{9}, 0, \frac{6}{9}, 0, 1, 132\right] \] The new tableau after pivoting on the 9 will be: $$\begin{bmatrix} \frac{2}{9} & 1 & \frac{1}{9} & 0 & 0 & 22 \\ \frac{61}{9} & 0 & -\frac{1}{9} & 1 & 0 & 727 \\ \frac{2}{9} & 0 & \frac{6}{9} & 0 & 1 & 132 \\ \end{bmatrix}$$ This tableau represents the updated solution. Would you like me to continue with parts (a) and (c) or provide additional details on interpreting this tableau? --- ### Follow-up Questions: 1. What are the feasible solutions of this tableau? 2. How do the group I and group II variables change after the pivot? 3. How do we determine if a solution is feasible in the simplex method? 4. What is the role of the pivot element in transforming the tableau? 5. How would another pivot on a different element affect the results? ### Tip: When performing a pivot, always ensure the chosen element is positive and that you correctly eliminate other entries in the pivot column.

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Math Problem Analysis

Mathematical Concepts

Linear Programming
Simplex Method
Tableau Pivoting

Formulas

Pivot operation formula: R_new = R - (pivot_row coefficient) * pivot_row
Conversion of row entries to make pivot 1: R_pivot = R_pivot / pivot element

Theorems

Simplex Algorithm
Feasibility and Optimization in Linear Programming

Suitable Grade Level

College Level

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