1. **State the problem:** We want to minimize the objective function $$Z = 5x_1 + 3x_2$$ subject to the constraints: $$2x_1 + 4x_2 \leq 12$$ $$2x_1 + 2x_2 = 10$$ $$5x_1 + 2x_2 \geq 10$$ $$x_1, x_2 \geq 0$$ 2. **Introduce slack, surplus, and artificial variables:** - For $$2x_1 + 4x_2 \leq 12$$, add slack variable $$s_1 \geq 0$$: $$2x_1 + 4x_2 + s_1 = 12$$ - For $$2x_1 + 2x_2 = 10$$, add artificial variable $$a_1 \geq 0$$: $$2x_1 + 2x_2 + a_1 = 10$$ - For $$5x_1 + 2x_2 \geq 10$$, subtract surplus variable $$s_2 \geq 0$$ and add artificial variable $$a_2 \geq 0$$: $$5x_1 + 2x_2 - s_2 + a_2 = 10$$ 3. **Formulate the Big-M objective function:** We penalize artificial variables with a large positive number $$M$$ to ensure they are driven out of the solution: $$Z = 5x_1 + 3x_2 + M a_1 + M a_2$$ 4. **Set up the initial simplex tableau with variables $$x_1, x_2, s_1, s_2, a_1, a_2$$ and solve using simplex method:** - The goal is to minimize $$Z$$ while satisfying all constraints. - Artificial variables $$a_1$$ and $$a_2$$ must be removed from the basis for a feasible solution. 5. **Summary:** - Use slack variable $$s_1$$ for the first inequality. - Use artificial variable $$a_1$$ for the equality. - Use surplus variable $$s_2$$ and artificial variable $$a_2$$ for the third inequality. - Minimize $$Z = 5x_1 + 3x_2 + M a_1 + M a_2$$. - Apply simplex iterations to find optimal $$x_1, x_2$$. Since the problem requires the Big-M method setup, the final answer is the formulation above ready for simplex iterations.