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 points:** Solve each pair of constraints as equalities to find intersection points. 4. **Find intersection of constraints 1 and 2:** $$4x_1 + 2.5x_2 = 60$$ $$2x_1 + 5x_2 = 60$$ Multiply second equation by 2: $$4x_1 + 10x_2 = 120$$ Subtract first equation: $$(4x_1 + 10x_2) - (4x_1 + 2.5x_2) = 120 - 60$$ $$7.5x_2 = 60$$ $$x_2 = \frac{60}{7.5} = 8$$ Substitute back: $$4x_1 + 2.5(8) = 60 \Rightarrow 4x_1 + 20 = 60 \Rightarrow 4x_1 = 40 \Rightarrow x_1 = 10$$ 5. **Find intersection of constraints 1 and 3:** $$4x_1 + 2.5x_2 = 60$$ $$5x_1 + 4x_2 = 82$$ Multiply first by 4: $$16x_1 + 10x_2 = 240$$ Multiply second by 2.5: $$12.5x_1 + 10x_2 = 205$$ Subtract second from first: $$(16x_1 + 10x_2) - (12.5x_1 + 10x_2) = 240 - 205$$ $$3.5x_1 = 35$$ $$x_1 = 10$$ Substitute back: $$4(10) + 2.5x_2 = 60 \Rightarrow 40 + 2.5x_2 = 60 \Rightarrow 2.5x_2 = 20 \Rightarrow x_2 = 8$$ 6. **Find intersection of constraints 2 and 3:** $$2x_1 + 5x_2 = 60$$ $$5x_1 + 4x_2 = 82$$ Multiply first by 4: $$8x_1 + 20x_2 = 240$$ Multiply second by 5: $$25x_1 + 20x_2 = 410$$ Subtract first from second: $$(25x_1 + 20x_2) - (8x_1 + 20x_2) = 410 - 240$$ $$17x_1 = 170$$ $$x_1 = 10$$ Substitute back: $$2(10) + 5x_2 = 60 \Rightarrow 20 + 5x_2 = 60 \Rightarrow 5x_2 = 40 \Rightarrow x_2 = 8$$ 7. **Check feasibility and objective value at intersection points:** All intersections give $$x_1 = 10, x_2 = 8$$. Calculate $$z$$: $$z = 5(10) + 4(8) = 50 + 32 = 82$$ 8. **Check corner points on axes:** - For $$x_1=0$$, check constraints: $$4(0) + 2.5x_2 \geq 60 \Rightarrow x_2 \geq 24$$ $$2(0) + 5x_2 \geq 60 \Rightarrow x_2 \geq 12$$ $$5(0) + 4x_2 \geq 82 \Rightarrow x_2 \geq 20.5$$ So $$x_2 \geq 24$$ to satisfy all. Calculate $$z = 5(0) + 4(24) = 96$$ - For $$x_2=0$$, check constraints: $$4x_1 + 0 \geq 60 \Rightarrow x_1 \geq 15$$ $$2x_1 + 0 \geq 60 \Rightarrow x_1 \geq 30$$ $$5x_1 + 0 \geq 82 \Rightarrow x_1 \geq 16.4$$ So $$x_1 \geq 30$$ to satisfy all. Calculate $$z = 5(30) + 4(0) = 150$$ 9. **Conclusion:** The minimum value of $$z$$ is $$82$$ at $$x_1 = 10, x_2 = 8$$. **Final answer:** $$\boxed{z_{min} = 82 \text{ at } (x_1, x_2) = (10, 8)}$$