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?