1. **Problem Statement:** We have two crops, Maize and Soybean, with constraints on land, labor, and nitrogen resources. We want to maximize returns. 2. **Define Variables:** Let $X_1$ = area (ha) for Maize Let $X_2$ = area (ha) for Soybean 3. **Formulate the Primal Problem:** Maximize returns: $$\text{Maximize } Z = 3600X_1 + 3000X_2$$ Subject to resource constraints: - Land: $$X_1 + X_2 \leq 5$$ - Labor: $$10X_1 + 25X_2 \leq 100$$ - Nitrogen: $$80X_1 + 25X_2 \leq 320$$ - Non-negativity: $$X_1 \geq 0, X_2 \geq 0$$ 4. **Formulate the Dual Problem:** Let $Y_1, Y_2, Y_3$ be the dual variables for land, labor, and nitrogen constraints respectively. Minimize: $$W = 5Y_1 + 100Y_2 + 320Y_3$$ Subject to: - For Maize: $$Y_1 + 10Y_2 + 80Y_3 \geq 3600$$ - For Soybean: $$Y_1 + 25Y_2 + 25Y_3 \geq 3000$$ - Non-negativity: $$Y_1, Y_2, Y_3 \geq 0$$ 5. **Graphic Solution for Primal:** Plot the constraints: - $$X_1 + X_2 = 5$$ - $$10X_1 + 25X_2 = 100$$ - $$80X_1 + 25X_2 = 320$$ Find feasible region in the first quadrant and evaluate $Z$ at vertices. 6. **Graphic Solution for Dual:** Plot inequalities: - $$Y_1 + 10Y_2 + 80Y_3 \geq 3600$$ - $$Y_1 + 25Y_2 + 25Y_3 \geq 3000$$ With $$Y_1, Y_2, Y_3 \geq 0$$ Find feasible region and minimize $W$. **Final answers:** - Primal: $$\max Z = 3600X_1 + 3000X_2$$ subject to $$X_1 + X_2 \leq 5$$ $$10X_1 + 25X_2 \leq 100$$ $$80X_1 + 25X_2 \leq 320$$ $$X_1, X_2 \geq 0$$ - Dual: $$\min W = 5Y_1 + 100Y_2 + 320Y_3$$ subject to $$Y_1 + 10Y_2 + 80Y_3 \geq 3600$$ $$Y_1 + 25Y_2 + 25Y_3 \geq 3000$$ $$Y_1, Y_2, Y_3 \geq 0$$