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This problem asks to solve a linear programming (LP) problem using the **simplex method**. Here's a summary of the task:

### Problem:
- **Objective**: Maximize \( z = 4x_1 + 6x_2 \)
- **Subject to constraints**:
  \[
  -x_1 + x_2 \leq 11
  \]
  \[
  -x_1 + x_2 \leq 27
  \]
  \[
  2x_1 + 5x_2 \leq 90
  \]
  \[
  x_1 \geq 0, \quad x_2 \geq 0
  \]

### Steps to solve using the simplex method:
1. **Convert inequalities to equalities by adding slack variables**:
   - Add slack variables \(s_1\), \(s_2\), and \(s_3\) for each inequality constraint to convert them into equalities.
   
   \[
   -x_1 + x_2 + s_1 = 11
   \]
   \[
   -x_1 + x_2 + s_2 = 27
   \]
   \[
   2x_1 + 5x_2 + s_3 = 90
   \]

2. **Set up the initial simplex tableau**:
   The tableau will include the objective function and the constraints with the slack variables.

3. **Iterate through the simplex method**:
   - Identify the entering variable (the variable with the most positive coefficient in the objective function row).
   - Identify the leaving variable (the row with the smallest positive ratio of the right-hand side to the entering variable’s coefficient).
   - Perform row operations to pivot and update the tableau.

4. **Repeat until optimal solution is found**.

Would you like me to set up the initial simplex tableau and walk you through each step in detail?

Use the simplex method to maximize z = 4x1 + 6x2 subject to -x1 + x2 <= 11, -x1 + x2 <= 27, 2x1 + 5x2 <= 90, with x1 >= 0 and x2 >= 0.

Solve a linear programming problem using the simplex method to maximize z = 4x1 + 6x2 subject to constraints. This problem introduces key optimization techniques and the simplex method.

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