This problem requires solving a linear programming (LP) problem using the Simplex Method. The goal is to maximize the objective function $z$ subject to given constraints.

Problem Summary:

• Objective Function:
  $\text{Maximize } z = 4x_1 + 3x_2 + 2x_3$

• Subject to Constraints:

  x_1 + x_2 + x_3 &\leq 5 \\ 2x_1 + x_2 - 2x_3 &\leq 8 \\ 3x_1 + 2x_2 + x_3 &\leq 10 \\ x_1, x_2 &\geq 0 \quad (\text{non-negativity constraints on } x_1 \text{ and } x_2) \end{aligned}$$

To solve this using the Simplex Method, we first convert the inequalities into equalities by introducing slack variables.

Step 1: Introducing Slack Variables

For each inequality, add a slack variable to transform it into an equation:

1. $x_1 + x_2 + x_3 + s_1 = 5$
2. $2x_1 + x_2 - 2x_3 + s_2 = 8$
3. $3x_1 + 2x_2 + x_3 + s_3 = 10$

Where $s_1, s_2, s_3 \geq 0$ are the slack variables.

Step 2: Set up the Initial Simplex Table

We now have the following system of equations: