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. **Understand the constraints:** These inequalities define the feasible region where the solution must lie. We will find the vertices of this region by solving the system of equations formed by the boundaries. 3. **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 $$x + y = 4$$. 4. **Find intersection points of pairs of lines:** - Intersection of $$x + 3y = 9$$ and $$2x + y = 8$$: Multiply second equation by 3: $$6x + 3y = 24$$. Subtract first equation: $$6x + 3y - (x + 3y) = 24 - 9$$ $$5x = 15 \Rightarrow x = 3$$. 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: $$x + 3y - (x + y) = 9 - 4$$ $$2y = 5 \Rightarrow y = 2.5$$. 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: $$2x + y - (x + y) = 8 - 4$$ $$x = 4$$. Substitute back: $$4 + y = 4 \Rightarrow y = 0$$. Point: $$(4, 0)$$. 5. **Check feasibility of points with all constraints including non-negativity:** - $$(3, 2)$$: Check $$x + y \geq 4$$: $$3 + 2 = 5 \geq 4$$ (ok), $$x \geq 0, y \geq 0$$ (ok). - $$(1.5, 2.5)$$: Check $$2x + 6y = 3 + 15 = 18 \geq 18$$ (ok), $$6x + 3y = 9 + 7.5 = 16.5 \leq 24$$ (ok), $$x + y = 4$$ (ok), non-negativity (ok). - $$(4, 0)$$: Check $$2(4) + 6(0) = 8 \geq 18$$ (no, fails), so discard. 6. **Check other boundary points:** - Check $$x=0$$ and $$y$$ from constraints: From $$x + y \geq 4$$, $$y \geq 4$$. From $$2x + 6y \geq 18$$, $$6y \geq 18 \Rightarrow y \geq 3$$. From $$6x + 3y \leq 24$$, $$3y \leq 24 \Rightarrow y \leq 8$$. So $$y$$ can be between 4 and 8 at $$x=0$$. Check point $$(0,4)$$: Objective: $$80(0) + 60(4) = 240$$. - Check $$y=0$$ and $$x$$ from constraints: From $$x + y \geq 4$$, $$x \geq 4$$. From $$2x + 6y \geq 18$$, $$2x \geq 18 \Rightarrow x \geq 9$$. From $$6x + 3y \leq 24$$, $$6x \leq 24 \Rightarrow x \leq 4$$. No $$x$$ satisfies both $$x \geq 9$$ and $$x \leq 4$$, so no feasible point at $$y=0$$. 7. **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 $$(0, 4)$$: $$0 + 240 = 240$$. 8. **Conclusion:** The minimum value of $$80x + 60y$$ subject to the constraints is $$240$$ at the point $$(0, 4)$$.