Question 1189862
Here's how to solve the given linear programming problem using the simplex method:

**1. Convert Inequalities to Equations:**

Introduce slack variables (s1, s2, s3) to convert the inequalities into equations:

*   x1 + x2 + s1 = 200
*   x2 + s2 = 125
*   3x1 + 6x2 + s3 = 900

**2. Set up the Initial Simplex Tableau:**

|   | x1 | x2 | s1 | s2 | s3 | RHS |
|---|---|---|---|---|---|---|
| Z | -8 | -16 | 0 | 0 | 0 | 0 |
|   | 1 | 1 | 1 | 0 | 0 | 200 |
|   | 0 | 1 | 0 | 1 | 0 | 125 |
|   | 3 | 6 | 0 | 0 | 1 | 900 |

**3. Identify the Pivot Column:**

The pivot column is the column with the most negative entry in the Z row. In this case, it's the x2 column (-16).

**4. Identify the Pivot Row:**

Divide the RHS values by the corresponding entries in the pivot column. The smallest non-negative result indicates the pivot row.

*   200 / 1 = 200
*   125 / 1 = 125
*   900 / 6 = 150

The pivot row is the second row.

**5. Pivot:**

The pivot element is the intersection of the pivot column and pivot row (1). Make the pivot element 1 (it already is) and all other elements in the pivot column 0.

|   | x1 | x2 | s1 | s2 | s3 | RHS |
|---|---|---|---|---|---|---|
| Z | -8 | 0 | 0 | 16 | 0 | 2000 |
|   | 1 | 0 | 1 | -1 | 0 | 75 |
|   | 0 | 1 | 0 | 1 | 0 | 125 |
|   | 3 | 0 | 0 | -6 | 1 | 150 |

**6. Repeat Steps 3-5:**

*   Pivot Column: x1 (-8)
*   Pivot Row: First Row (75/1 is the smallest)
*   Pivot:

|   | x1 | x2 | s1 | s2 | s3 | RHS |
|---|---|---|---|---|---|---|
| Z | 0 | 0 | 8 | 8 | 0 | 2600 |
|   | 1 | 0 | 1 | -1 | 0 | 75 |
|   | 0 | 1 | 0 | 1 | 0 | 125 |
|   | 0 | 0 | -3 | -3 | 1 | -75 |

**7. Optimal Solution:**

Since there are no more negative entries in the Z row, the optimal solution is reached.

*   x1 = 50
*   x2 = 125
*   Z = 2400

Therefore, the maximum value of Z is 2400 when x1 = 50 and x2 = 125.