1. **State the problem:** Minimize the objective function $$80x + 60y$$ subject to the constraints: $$2x + 6y \geq 18$$ $$6x + 3y \leq 24$$ $$x + y \geq 4$$ $$x \geq 0, y \geq 0$$ 2. **Rewrite inequalities as equalities to find intersection points:** - From $$2x + 6y = 18$$ simplify to $$x + 3y = 9$$ - From $$6x + 3y = 24$$ simplify to $$2x + y = 8$$ - From $$x + y = 4$$ 3. **Find intersection points of pairs of lines:** - Intersection of $$x + 3y = 9$$ and $$2x + y = 8$$: Multiply second by 3: $$6x + 3y = 24$$ Subtract first: $$6x + 3y - (x + 3y) = 24 - 9 \Rightarrow 5x = 15 \Rightarrow x = 3$$ Substitute back: $$3 + 3y = 9 \Rightarrow 3y = 6 \Rightarrow y = 2$$ Point: $$(3, 2)$$ - Intersection of $$x + 3y = 9$$ and $$x + y = 4$$: Subtract second from first: $$x + 3y - (x + y) = 9 - 4 \Rightarrow 2y = 5 \Rightarrow y = 2.5$$ Substitute back: $$x + 2.5 = 4 \Rightarrow x = 1.5$$ Point: $$(1.5, 2.5)$$ - Intersection of $$2x + y = 8$$ and $$x + y = 4$$: Subtract second from first: $$2x + y - (x + y) = 8 - 4 \Rightarrow x = 4$$ Substitute back: $$4 + y = 4 \Rightarrow y = 0$$ Point: $$(4, 0)$$ 4. **Check feasibility of points with all constraints:** - $$(3, 2)$$ satisfies all constraints. - $$(1.5, 2.5)$$ satisfies all constraints. - $$(4, 0)$$ fails $$2x + 6y \geq 18$$ since $$8 \not\geq 18$$, discard. 5. **Check boundary points:** - At $$x=0$$, from constraints $$y \geq 4$$ and $$6y \geq 18 \Rightarrow y \geq 3$$, and $$3y \leq 24 \Rightarrow y \leq 8$$, so $$y$$ can be between 4 and 8. Check point $$(0, 4)$$: feasible. - At $$y=0$$, constraints require $$x \geq 9$$ and $$x \leq 4$$, no feasible $$x$$. 6. **Evaluate objective function at feasible vertices:** - At $$(3, 2)$$: $$80(3) + 60(2) = 240 + 120 = 360$$ - At $$(1.5, 2.5)$$: $$80(1.5) + 60(2.5) = 120 + 150 = 270$$ - At $$(0, 4)$$: $$80(0) + 60(4) = 0 + 240 = 240$$ 7. **Conclusion:** The minimum value of $$80x + 60y$$ subject to the constraints is $$240$$ at the point $$(0, 4)$$.