To determine the location of the pivot element in the simplex tableau, we need to follow these steps:
### Step 1: Identify the Pivot Column
The pivot column is determined by finding the most negative value in the objective function row (the last row of the tableau, excluding the rightmost column).
The last row is:
[tex]\[
-93 \quad -67 \quad 0 \quad 0 \quad 1
\][/tex]
The most negative value among these is [tex]\(-93\)[/tex]. Thus, the pivot column is the column corresponding to [tex]\(-93\)[/tex], which is column 1.
### Step 2: Identify the Pivot Row
To find the pivot row, we calculate the ratio of the values in the rightmost column to the corresponding values in the identified pivot column. We select the row with the smallest non-negative ratio.
Here's the first two rows (excluding the objective function row):
[tex]\[
\begin{array}{cccc|c}
3 & 2 & 1 & 0 & 50 \\
5 & 4 & 0 & 1 & 90 \\
\end{array}
\][/tex]
We'll calculate the ratios:
- For row 1: [tex]\[\frac{50}{3} \approx 16.67\][/tex]
- For row 2: [tex]\[\frac{90}{5} = 18\][/tex]
The smallest non-negative ratio is [tex]\(\approx 16.67\)[/tex], which corresponds to row 1.
### Conclusion
The pivot element is located at the intersection of row 1 and column 1.
Thus, the pivot element is located in:
- Row: [tex]\(0\)[/tex]
- Column: [tex]\(0\)[/tex]
So, the final answer is:
The pivot element is located in row [tex]\(0\)[/tex], column [tex]\(0\)[/tex].
