1. **Problem Statement:** Minimize the objective function $$Z = 5x_1 + 4x_2$$ subject to the constraints: $$4x_1 + x_2 \geq 40$$ $$2x_1 + 3x_2 \geq 90$$ $$x_1 \geq 0, x_2 \geq 0$$ 2. **Understanding the problem:** We want to find values of $x_1$ and $x_2$ that satisfy the inequalities and minimize $Z$. The graphical method involves plotting the constraints and finding the feasible region. 3. **Rewrite constraints as equalities to plot boundary lines:** - Line 1: $$4x_1 + x_2 = 40$$ - Line 2: $$2x_1 + 3x_2 = 90$$ 4. **Find intercepts for Line 1:** - When $x_1=0$, $x_2=40$ - When $x_2=0$, $x_1=10$ 5. **Find intercepts for Line 2:** - When $x_1=0$, $3x_2=90 \Rightarrow x_2=30$ - When $x_2=0$, $2x_1=90 \Rightarrow x_1=45$ 6. **Plot these lines and identify the feasible region:** The feasible region is where both inequalities hold (above or on the lines) and $x_1,x_2 \geq 0$. 7. **Find corner points of the feasible region:** - Intersection of Line 1 and Line 2: Solve system: $$4x_1 + x_2 = 40$$ $$2x_1 + 3x_2 = 90$$ Multiply first by 3: $$12x_1 + 3x_2 = 120$$ Subtract second: $$(12x_1 + 3x_2) - (2x_1 + 3x_2) = 120 - 90$$ $$10x_1 = 30 \Rightarrow x_1 = 3$$ Substitute back: $$4(3) + x_2 = 40 \Rightarrow 12 + x_2 = 40 \Rightarrow x_2 = 28$$ - Intercepts on axes (check feasibility): At $x_1=0$, $x_2$ must satisfy both inequalities: $$4(0) + x_2 \geq 40 \Rightarrow x_2 \geq 40$$ $$2(0) + 3x_2 \geq 90 \Rightarrow 3x_2 \geq 90 \Rightarrow x_2 \geq 30$$ So $x_2 \geq 40$ to satisfy both. At $x_2=0$, $x_1$ must satisfy: $$4x_1 + 0 \geq 40 \Rightarrow x_1 \geq 10$$ $$2x_1 + 0 \geq 90 \Rightarrow x_1 \geq 45$$ So $x_1 \geq 45$ to satisfy both. 8. **Evaluate $Z$ at corner points:** - At $(3,28)$: $$Z = 5(3) + 4(28) = 15 + 112 = 127$$ - At $(0,40)$: $$Z = 5(0) + 4(40) = 160$$ - At $(45,0)$: $$Z = 5(45) + 4(0) = 225$$ 9. **Conclusion:** Minimum value of $Z$ is 127 at $x_1=3$, $x_2=28$. --- **Slug:** linear_programming **Subject:** optimization