1. **State the problem:** Minimize the objective function $$Z = 4x + 3y$$ subject to the constraints: $$2x + y \geq 10$$ $$3x + 2y \leq 6$$ $$x + y \geq 6$$ $$x, y \geq 0$$ 2. **Convert inequalities to equalities using slack, surplus, and artificial variables:** For $$2x + y \geq 10$$, rewrite as $$2x + y - s_1 = 10$$ with surplus variable $$s_1 \geq 0$$ and add artificial variable $$a_1$$: $$2x + y - s_1 + a_1 = 10$$ For $$3x + 2y \leq 6$$, add slack variable $$s_2$$: $$3x + 2y + s_2 = 6$$ For $$x + y \geq 6$$, rewrite as $$x + y - s_3 = 6$$ with surplus variable $$s_3 \geq 0$$ and add artificial variable $$a_2$$: $$x + y - s_3 + a_2 = 6$$ 3. **Set up the Big-M objective function:** Introduce a large positive number $$M$$ and penalize artificial variables: $$Z = 4x + 3y + M a_1 + M a_2$$ 4. **Formulate the initial simplex tableau with variables $$x, y, s_1, s_2, s_3, a_1, a_2$$ and constraints:** $$\begin{cases} 2x + y - s_1 + a_1 = 10 \\ 3x + 2y + s_2 = 6 \\ x + y - s_3 + a_2 = 6 \end{cases}$$ 5. **Apply the simplex method to minimize $$Z$$:** - Start with artificial variables $$a_1$$ and $$a_2$$ in the basis. - Perform pivot operations to remove artificial variables from the basis. - Continue iterations until no negative coefficients remain in the objective row. 6. **Solve the system or use software to find the optimal solution:** After applying the Big-M method and simplex iterations, the optimal solution is: $$x = 0, y = 3, Z = 4(0) + 3(3) = 9$$ 7. **Verify constraints:** - $$2(0) + 3 = 3 \geq 10$$? No, so this solution is infeasible. Re-examining constraints and calculations, the problem is infeasible because constraints $$2x + y \geq 10$$ and $$3x + 2y \leq 6$$ cannot be satisfied simultaneously with $$x, y \geq 0$$. **Final conclusion:** The problem has no feasible solution under the given constraints.