1. **State the problem:** We need to find how many chairs of type A and type B can be produced given the cutting and assembly time constraints. 2. **Define variables:** Let $x$ be the number of Chair A produced. Let $y$ be the number of Chair B produced. 3. **Write the constraints as equations:** Cutting time: $2x + y \leq 40$ Assembly time: $x + 2y \leq 35$ 4. **Explain the constraints:** - Each Chair A requires 2 hours cutting and 1 hour assembly. - Each Chair B requires 1 hour cutting and 2 hours assembly. - Total cutting time available is 40 hours. - Total assembly time available is 35 hours. 5. **Find the maximum number of chairs that can be produced:** We want to find integer solutions $(x,y)$ satisfying both inequalities. 6. **Solve the system by finding intersection points:** From cutting time: $2x + y = 40 \Rightarrow y = 40 - 2x$ From assembly time: $x + 2y = 35 \Rightarrow x = 35 - 2y$ Substitute $y$ from first into second: $$x = 35 - 2(40 - 2x) = 35 - 80 + 4x = -45 + 4x$$ Rearranged: $$x - 4x = -45 \Rightarrow -3x = -45 \Rightarrow x = 15$$ Find $y$: $$y = 40 - 2(15) = 40 - 30 = 10$$ 7. **Check constraints:** Cutting: $2(15) + 10 = 30 + 10 = 40 \leq 40$ (ok) Assembly: $15 + 2(10) = 15 + 20 = 35 \leq 35$ (ok) 8. **Interpretation:** The factory can produce 15 chairs of type A and 10 chairs of type B using all available cutting and assembly time. **Final answer:** $$\boxed{x=15, y=10}$$