1. **State the problem:** We want to find the minimum and maximum values of the objective function $$z = 5x + 4y$$ subject to the constraints: $$2x + y \leq 20$$ $$10x + y \geq 36$$ $$2x + 5y \geq 36$$ 2. **Understand the constraints:** These inequalities define a feasible region on the coordinate plane where all conditions are satisfied simultaneously. 3. **Find the intersection points (vertices) of the feasible region:** The minimum and maximum values of a linear objective function over a polygonal feasible region occur at vertices. - Solve intersections of pairs of lines: (a) Intersection of $$2x + y = 20$$ and $$10x + y = 36$$: Subtract first from second: $$10x + y - (2x + y) = 36 - 20 \Rightarrow 8x = 16 \Rightarrow x = 2$$ Substitute back: $$2(2) + y = 20 \Rightarrow 4 + y = 20 \Rightarrow y = 16$$ Vertex 1: $$(2,16)$$ (b) Intersection of $$2x + y = 20$$ and $$2x + 5y = 36$$: From first: $$y = 20 - 2x$$ Substitute into second: $$2x + 5(20 - 2x) = 36 \Rightarrow 2x + 100 - 10x = 36 \Rightarrow -8x = -64 \Rightarrow x = 8$$ Then: $$y = 20 - 2(8) = 20 - 16 = 4$$ Vertex 2: $$(8,4)$$ (c) Intersection of $$10x + y = 36$$ and $$2x + 5y = 36$$: From first: $$y = 36 - 10x$$ Substitute into second: $$2x + 5(36 - 10x) = 36 \Rightarrow 2x + 180 - 50x = 36 \Rightarrow -48x = -144 \Rightarrow x = 3$$ Then: $$y = 36 - 10(3) = 36 - 30 = 6$$ Vertex 3: $$(3,6)$$ 4. **Check which vertices satisfy all inequalities:** - Vertex (2,16): - $$2(2)+16=20 \leq 20$$ ✓ - $$10(2)+16=36 \geq 36$$ ✓ - $$2(2)+5(16)=4+80=84 \geq 36$$ ✓ - Vertex (8,4): - $$2(8)+4=20 \leq 20$$ ✓ - $$10(8)+4=84 \geq 36$$ ✓ - $$2(8)+5(4)=16+20=36 \geq 36$$ ✓ - Vertex (3,6): - $$2(3)+6=12 \leq 20$$ ✓ - $$10(3)+6=36 \geq 36$$ ✓ - $$2(3)+5(6)=6+30=36 \geq 36$$ ✓ All three vertices are feasible. 5. **Evaluate the objective function at each vertex:** - At (2,16): $$z = 5(2) + 4(16) = 10 + 64 = 74$$ - At (8,4): $$z = 5(8) + 4(4) = 40 + 16 = 56$$ - At (3,6): $$z = 5(3) + 4(6) = 15 + 24 = 39$$ 6. **Determine minimum and maximum:** - Minimum value of $$z$$ is $$39$$ at $$(3,6)$$ - Maximum value of $$z$$ is $$74$$ at $$(2,16)$$ **Final answer:** Minimum $$z = 39$$ at $$(3,6)$$ Maximum $$z = 74$$ at $$(2,16)$$