To find the pivot in the simplex tableau, follow these steps:

Step 1: Identify the Pivot Column

The pivot column is determined by the most negative value in the bottom row (objective function row).
Looking at the last row:
$[0, -3, -1, 1, 0, 1]$
The most negative value is $-3$, so Column 2 (corresponding to $x_2$) is the pivot column.

Step 2: Identify the Pivot Row

Divide the right-hand side (RHS) values by the corresponding positive entries in the pivot column:

• Row 1: $\frac{15}{7} = 2.14$
• Row 2: $\frac{10}{6} = 1.67$
  Since $1.67$ is the smallest positive ratio, Row 2 is the pivot row.

Step 3: Conclusion

The pivot element is the intersection of Row 2 and Column 2, which is 6.

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Final Answer:

• The Pivot is: 6
• Located in Row: 2
• Located in Column: 2

Would you like more details or have any questions?

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Related Questions:

1. How do you handle ties when selecting a pivot row?
2. What happens if all entries in the pivot column are non-positive?
3. How does the simplex method proceed after selecting the pivot?
4. Can there be multiple optimal solutions in a simplex tableau?
5. What is the role of artificial variables in the simplex method?

Tip: Always choose the pivot column using the most negative value in the objective row to maximize or minimize the objective function.