1. **State the problem:** Minimize the objective function $$z = 5x_1 + 4x_2$$ subject to the constraints: $$4x_1 + 2.5x_2 \geq 60$$ $$2x_1 + 5x_2 \geq 60$$ $$5x_1 + 4x_2 \geq 82$$ with $$x_1, x_2 \geq 0$$. 2. **Rewrite constraints for clarity:** We want to find $$x_1, x_2$$ that satisfy all inequalities and minimize $$z$$. 3. **Convert inequalities to equalities to find boundary lines:** $$4x_1 + 2.5x_2 = 60$$ $$2x_1 + 5x_2 = 60$$ $$5x_1 + 4x_2 = 82$$ 4. **Find intersection points of these lines (feasible region vertices):** - Intersection of first and second: Solve system: $$4x_1 + 2.5x_2 = 60$$ $$2x_1 + 5x_2 = 60$$ Multiply second by 2: $$4x_1 + 10x_2 = 120$$ Subtract first: $$(4x_1 + 10x_2) - (4x_1 + 2.5x_2) = 120 - 60$$ $$7.5x_2 = 60 \Rightarrow x_2 = 8$$ Plug back: $$4x_1 + 2.5(8) = 60 \Rightarrow 4x_1 + 20 = 60 \Rightarrow 4x_1 = 40 \Rightarrow x_1 = 10$$ So point A: $$(10, 8)$$ - Intersection of first and third: $$4x_1 + 2.5x_2 = 60$$ $$5x_1 + 4x_2 = 82$$ Multiply first by 4: $$16x_1 + 10x_2 = 240$$ Multiply second by 2.5: $$12.5x_1 + 10x_2 = 205$$ Subtract second from first: $$(16x_1 + 10x_2) - (12.5x_1 + 10x_2) = 240 - 205$$ $$3.5x_1 = 35 \Rightarrow x_1 = 10$$ Plug back: $$4(10) + 2.5x_2 = 60 \Rightarrow 40 + 2.5x_2 = 60 \Rightarrow 2.5x_2 = 20 \Rightarrow x_2 = 8$$ So point B: $$(10, 8)$$ (same as A) - Intersection of second and third: $$2x_1 + 5x_2 = 60$$ $$5x_1 + 4x_2 = 82$$ Multiply first by 4: $$8x_1 + 20x_2 = 240$$ Multiply second by 5: $$25x_1 + 20x_2 = 410$$ Subtract first from second: $$(25x_1 + 20x_2) - (8x_1 + 20x_2) = 410 - 240$$ $$17x_1 = 170 \Rightarrow x_1 = 10$$ Plug back: $$2(10) + 5x_2 = 60 \Rightarrow 20 + 5x_2 = 60 \Rightarrow 5x_2 = 40 \Rightarrow x_2 = 8$$ So point C: $$(10, 8)$$ (same as A and B) 5. **Check feasibility and objective value at point (10,8):** All constraints are satisfied. Calculate $$z$$: $$z = 5(10) + 4(8) = 50 + 32 = 82$$ 6. **Check intercepts to ensure no other feasible points with lower $$z$$:** - At $$x_1=0$$: From $$4x_1 + 2.5x_2 \geq 60$$, $$2.5x_2 \geq 60 \Rightarrow x_2 \geq 24$$ Calculate $$z = 5(0) + 4(24) = 96$$ - At $$x_2=0$$: From $$4x_1 + 2.5x_2 \geq 60$$, $$4x_1 \geq 60 \Rightarrow x_1 \geq 15$$ Calculate $$z = 5(15) + 4(0) = 75$$ 7. **Conclusion:** The minimum value of $$z$$ is $$82$$ at $$x_1=10, x_2=8$$. **Final answer:** $$\boxed{z_{min} = 82 \text{ at } (x_1, x_2) = (10, 8)}$$