Math Problem Statement

pivot the 2 to create a new tableau and obtain its corresponding solutions in bracket form

Pivot the tableau about 2, 9, 7, and 1 (row 2, column 2) and compute the solution to the resulting tableau.

Solution

Let's address the question step-by-step. Here's the approach:

Step 1: Identify Group I and Group II Variables (part (a))

  • Group I variables are the basic variables in the tableau, i.e., the variables associated with columns containing only one "1" and zeros elsewhere.
  • Group II variables are the non-basic variables, which don't have this "one-in-column" property.

From the given tableau:

x & y & u & v & M \\ 2 & 9 & 1 & 0 & 0 & 198 \\ 7 & 1 & 0 & 1 & 0 & 749 \\ -10 & -6 & 0 & 0 & 1 & 0 \\ \end{array}$$ - **Group I**: Variables associated with pivot columns, which are **u** and **v**. - **Group II**: Non-basic variables, which are **x, y,** and **M**. ### Step 2: Perform the Pivot Operation (part (b)) We're asked to pivot around 2 (the entry in row 1, column 1 for **x**). To perform the pivot: 1. Divide the first row by 2 to make the pivot element 1. 2. Adjust the other rows so that they have a zero in the **x** column. Let's apply this: 1. **Divide Row 1 by 2**: $$\left[ 1, \frac{9}{2}, \frac{1}{2}, 0, 0 \middle| \frac{198}{2} = 99 \right]$$ Row 1 becomes: $$\left[ 1, \frac{9}{2}, \frac{1}{2}, 0, 0 \middle| 99 \right]$$ 2. **Row Operations for Zeroing Out Column 1 in Other Rows**: - **Row 2**: $$\text{Row 2} - 7 \times \text{Row 1}$$ $$\left[ 0, -\frac{61}{2}, -\frac{7}{2}, 1, 0 \middle| 52 \right]$$ - **Row 3**: $$\text{Row 3} + 10 \times \text{Row 1}$$ $$\left[ 0, 39, 5, 0, 1 \middle| 990 \right]$$ The new tableau becomes: $$\begin{array}{cccccc|c} x & y & u & v & M \\ 1 & \frac{9}{2} & \frac{1}{2} & 0 & 0 & 99 \\ 0 & -\frac{61}{2} & -\frac{7}{2} & 1 & 0 & 52 \\ 0 & 39 & 5 & 0 & 1 & 990 \\ \end{array}$$ ### Step 3: Interpret the Solution The solution for the tableau corresponds to the values of basic variables in terms of the values in the last column: - **x = 99** - **v = 52** - **M = 990** ### Summary of Results in Bracket Form $$(x, y, u, v, M) = (99, 0, 0, 52, 990)$$ Would you like a more detailed explanation, or do you have any specific questions? --- ### Related Questions 1. How would you pivot around a different element in the tableau? 2. How does the simplex algorithm choose the pivot element in general? 3. Can the solution change if we choose a different pivot element? 4. How can we determine if a tableau represents an optimal solution? 5. Why does increasing the value of $$M$$ improve the objective function in linear programming? ### Tip: When performing pivots, ensure you consistently apply row operations to maintain feasibility and accuracy in simplex calculations.

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Math Problem Analysis

Mathematical Concepts

Linear Programming
Simplex Method
Pivoting in Tableau

Formulas

Pivoting operations in simplex tableau
Row operations for tableau transformations

Theorems

Fundamental Theorem of Linear Programming

Suitable Grade Level

Undergraduate Level