To determine the pivot element in the simplex tableau, follow these steps:

Step 1: Identify the entering variable.

The entering variable corresponds to the most negative coefficient in the $z$-row (the last row of the tableau). This coefficient is the one that maximizes the increase in $z$ when it becomes positive.

In this tableau, the $z$-row is: $-5, -8, 0, 0, 0, 1$ The most negative value is $-8$, which corresponds to $x_2$. Thus, $x_2$ will enter the basis.

Step 2: Identify the departing variable (determine the pivot row).

To identify the pivot row, divide each entry in the right-hand side (RHS) column by the corresponding positive entry in the $x_2$ column. This calculation is done for rows with positive entries in the $x_2$ column only.

The right-hand side column is: $7, 8, 10$ and the entries in the $x_2$ column are: $1, 4, 5$

Calculating the ratios:

• Row 1: $\frac{7}{1} = 7$
• Row 2: $\frac{8}{4} = 2$
• Row 3: $\frac{10}{5} = 2$

The smallest ratio is 2, and it occurs in both Row 2 and Row 3. According to the problem's instructions, we choose the row with the smaller row number, which is Row 2.

Step 3: Determine the pivot element.

The pivot element is in Row 2 and corresponds to $x_2$ in the second column. Therefore, the pivot element is:

$4$

Answer:

The pivot element is 4 in Row 2, Column 2.

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Would you like a further breakdown of these steps, or have additional questions?

Related Questions

1. How do you choose the entering variable in a simplex tableau?
2. What happens if two or more ratios are the same when determining the pivot row?
3. How does pivoting affect the simplex tableau in the next iteration?
4. What if there are no positive entries in a column when determining the departing variable?
5. How do we handle degeneracy in the simplex method?

Tip

In simplex tableau, always remember to check the entire $z$-row to confirm the most negative value before choosing the entering variable.