Questions: Pivot once as indicated in the given simplex x1 x2 x3 s1 s2 z tableau. Read the solution from the result. [1 8 2 1 0 0 56; 2 4 1 0 1 0 18; -1 -6 -2 0 0 1 0] Pivot around the highlighted entry. [x1 x2 x3 s1 s2 z; (blank) 0 (blank) 1 (blank) 0 (blank); 2 4 1 0 1 0 18; (blank) 0 (blank) 0 (blank) 1 (blank)] (Simplify your answers.)

Solution Steps

To solve the given simplex tableau problem, we need to perform a pivot operation around the highlighted entry (4 in the second row and second column). The steps are as follows:

1. Identify the pivot element (4 in this case).
2. Divide the entire pivot row by the pivot element to make the pivot element 1.
3. Adjust the other rows to make the rest of the column containing the pivot element zero.
4. Read the solution from the resulting tableau.

The initial simplex tableau is given as follows:

\[ \begin{array}{cccccc|c} x_1 & x_2 & x_3 & s_1 & s_2 & z \\ \hline 1 & 8 & 2 & 1 & 0 & 0 & 56 \\ 2 & 4 & 1 & 0 & 1 & 0 & 18 \\ \hline -1 & -6 & -2 & 0 & 0 & 1 & 0 \end{array} \]

We perform a pivot operation around the highlighted entry \(4\) located at row \(1\) and column \(1\). The pivot element is \(4\).

1. Divide the pivot row by \(4\):

\[ \text{New Row 1} = \frac{1}{4} \times \text{Row 1} = [0, 1, 0, 0, 0, 0, 4] \]

2. Adjust the other rows to make the rest of the column containing the pivot element zero:

• For Row 0: \[ \text{New Row 0} = \text{Row 0} - 8 \times \text{New Row 1} = [1, 0, 2, 1, 0, 0, 24] \]

• For Row 2: \[ \text{New Row 2} = \text{Row 2} + 6 \times \text{New Row 1} = [-1, 0, -2, 0, 0, 1, 24] \]

The resulting tableau after the pivot operation is:

\[ \begin{array}{cccccc|c} x_1 & x_2 & x_3 & s_1 & s_2 & z \\ \hline 1 & 0 & 2 & 1 & 0 & 0 & 24 \\ 0 & 1 & 0 & 0 & 0 & 0 & 4 \\ \hline -1 & 0 & -2 & 0 & 0 & 1 & 24 \end{array} \]

From the final tableau, we can read the solution:

• \(x_1 = 0\)
• \(x_2 = 4\)
• \(x_3 = 0\)
• \(s_1 = 24\) (slack variable)
• \(s_2 = 0\) (slack variable)
• The value of the objective function \(z = 24\)

Final Answer