1. **State the problem:** We want to determine how many posters ($x$) and flyers ($y$) the printing shop can produce given time constraints for designing and printing, and also calculate the earnings. 2. **Define variables:** Let $x$ = number of posters, $y$ = number of flyers. 3. **Write the constraints:** - Design time: Each poster requires 4 minutes, each flyer 2 minutes, total design time available is 480 minutes. $$4x + 2y \leq 480$$ - Printing time: Each poster requires 2 minutes, each flyer 1 minute, total printing time available is 300 minutes. $$2x + y \leq 300$$ 4. **Objective function (earnings):** Each poster earns 35, each flyer earns 15. $$\text{Earnings} = 35x + 15y$$ 5. **Explain the problem type:** This is a linear programming problem where we want to maximize earnings subject to time constraints. 6. **Find intercepts for constraints:** - For design time: If $y=0$, then $4x=480 \Rightarrow x=120$; if $x=0$, then $2y=480 \Rightarrow y=240$. - For printing time: If $y=0$, then $2x=300 \Rightarrow x=150$; if $x=0$, then $y=300$. 7. **Check corner points of feasible region:** - Point A: $(0,0)$ - Point B: $(0,240)$ from design constraint - Point C: $(120,0)$ from design constraint - Point D: $(0,300)$ from printing constraint (not feasible since design constraint violated) - Point E: $(150,0)$ from printing constraint (not feasible since design constraint violated) 8. **Find intersection of constraints:** Solve system: $$4x + 2y = 480$$ $$2x + y = 300$$ Multiply second equation by 2: $$4x + 2y = 600$$ Subtract first equation: $$(4x + 2y) - (4x + 2y) = 600 - 480 \Rightarrow 0 = 120$$ This is a contradiction, so no intersection; the feasible region is bounded by the more restrictive constraints. 9. **Check which constraint is more restrictive:** For $x=0$, design allows $y=240$, printing allows $y=300$; design is more restrictive. For $y=0$, design allows $x=120$, printing allows $x=150$; design is more restrictive. 10. **Evaluate earnings at corner points:** - At $(0,0)$: $35(0) + 15(0) = 0$ - At $(0,240)$: $35(0) + 15(240) = 3600$ - At $(120,0)$: $35(120) + 15(0) = 4200$ 11. **Conclusion:** Maximum earnings occur at producing 120 posters and 0 flyers, earning 4200.