1. **State the problem:** We need to find the equilibrium prices $P_A$ and $P_B$ and quantities $Q_A$ and $Q_B$ where demand equals supply for both products A and B. 2. **Write down the given functions:** Demand: $$Q_A^d = 40 - 2P_A + P_B$$ $$Q_B^d = 300 - P_B + P_A$$ Supply: $$Q_A^s = 10 + P_A$$ $$Q_B^s = 5 + 2P_B$$ 3. **Equilibrium condition:** At equilibrium, quantity demanded equals quantity supplied for each product: $$Q_A^d = Q_A^s$$ $$Q_B^d = Q_B^s$$ 4. **Set up the system of equations:** $$40 - 2P_A + P_B = 10 + P_A$$ $$300 - P_B + P_A = 5 + 2P_B$$ 5. **Simplify each equation:** For product A: $$40 - 2P_A + P_B = 10 + P_A \implies 40 - 10 + P_B = 2P_A + P_A \implies 30 + P_B = 3P_A$$ Rearranged: $$3P_A - P_B = 30$$ For product B: $$300 - P_B + P_A = 5 + 2P_B \implies 300 - 5 + P_A = 2P_B + P_B \implies 295 + P_A = 3P_B$$ Rearranged: $$P_A - 3P_B = -295$$ 6. **Solve the system:** From the first equation: $$3P_A - P_B = 30 \implies P_B = 3P_A - 30$$ Substitute into the second: $$P_A - 3(3P_A - 30) = -295$$ $$P_A - 9P_A + 90 = -295$$ $$-8P_A = -385$$ $$P_A = \frac{385}{8} = 48.125$$ Find $P_B$: $$P_B = 3(48.125) - 30 = 144.375 - 30 = 114.375$$ 7. **Find equilibrium quantities:** $$Q_A = Q_A^s = 10 + P_A = 10 + 48.125 = 58.125$$ $$Q_B = Q_B^s = 5 + 2P_B = 5 + 2(114.375) = 5 + 228.75 = 233.75$$ **Final answer:** $$P_A = 48.125, \quad Q_A = 58.125$$ $$P_B = 114.375, \quad Q_B = 233.75$$