To solve the given integer programming problem using the Gomory's cutting plane method, we start with the linear programming relaxation of the problem. The problem is to maximize $z = x_1 - x_2$ subject to the constraints:

1. $x_1 + x_2 \leq 4$
2. $6x_1 + 2x_2 \leq 9$
3. $x_1, x_2 \geq 0$
4. $x_1, x_2 \in \mathbb{Z}$

Step-by-Step Solution:

1. Linear Programming Relaxation:

First, we solve the linear programming relaxation by ignoring the integer constraints. This means we solve:

\text{Maximize} & \quad z = x_1 - x_2 \\ \text{subject to} & \quad x_1 + x_2 \leq 4 \\ & \quad 6x_1 + 2x_2 \leq 9 \\ & \quad x_1, x_2 \geq 0 \end{aligned}$$ 2. **Solve the Linear Programming Relaxation:** We can use the Simplex method to solve this. Let's set up the Simplex tableau. To convert the inequalities to equations, we introduce slack variables $$s_1$$ and $$s_2$$: $$\begin{aligned} x_1 + x_2 + s_1 &= 4 \\ 6x_1 + 2x_2 + s_2 &= 9 \\ \end{aligned}$$ Our initial tableau is: $$\begin{array}{c|cccccc|c} \text{Basis} & x_1 & x_2 & s_1 & s_2 & \text{RHS} \\ \hline s_1 & 1 & 1 & 1 & 0 & 4 \\ s_2 & 6 & 2 & 0 & 1 & 9 \\ \hline z & -1 & 1 & 0 & 0 & 0 \\ \end{array}$$ **Simplex Iteration:** $$\begin{array}{c|cccc|c} & x_1 & x_2 & s_1 & s_2 & \text{RHS} \\ \hline s_1 & 1 & 1 & 1 & 0 & 4 \\ s_2 & 6 & 2 & 0 & 1 & 9 \\ \hline z & -1 & 1 & 0 & 0 & 0 \\ \end{array}$$ **Choosing the pivot column (entering variable):** $$x_1$$ (most negative coefficient in the $$z$$-row) **Choosing the pivot row (leaving variable):** $$\frac{\text{RHS}}{\text{Pivot Column}} = \min \left( \frac{4}{1}, \frac{9}{6} \right) = \frac{9}{6} = 1.5 \implies \text{row 2 is the pivot row}$$ Pivot element is $$6$$. Performing the row operations to get the new tableau: $$\begin{array}{c|cccc|c} & x_1 & x_2 & s_1 & s_2 & \text{RHS} \\ \hline s_1 & \frac{1}{6} & \frac{2}{6} & 0 & \frac{1}{6} & \frac{3}{2} \\ x_1 & 1 & \frac{1}{3} & 0 & \frac{1}{6} & \frac{3}{2} \\ \hline z & 0 & \frac{4}{6} & 0 & \frac{1}{6} & \frac{3}{2} \\ \end{array}$$ **Next Iteration:** Pivot element is now $$\frac{2}{6}$$ in row 1. After further iterations, you would eventually find the optimal solution of the LP relaxation. 3. **Introduce Gomory's Cutting Plane:** Once the optimal solution of the relaxed problem is found, if it is not integer, we introduce Gomory’s fractional cuts.