1. **State the problem:** We are given demand and supply functions for two commodities and need to find the equilibrium prices $p_1^*$, $p_2^*$ and quantities $Q_1^*$, $Q_2^*$ where demand equals supply for each commodity. 2. **Write down the given functions:** $$Q_{d1} = 18 - 3P_1 + P_2$$ $$Q_{d2} = 12 + P_1 - 2P_2$$ $$Q_{s1} = -2 + 4P_1$$ $$Q_{s2} = -2 + 3P_2$$ 3. **Set demand equal to supply for each commodity to find equilibrium:** $$Q_{d1} = Q_{s1} \implies 18 - 3P_1 + P_2 = -2 + 4P_1$$ $$Q_{d2} = Q_{s2} \implies 12 + P_1 - 2P_2 = -2 + 3P_2$$ 4. **Simplify each equation:** For commodity 1: $$18 - 3P_1 + P_2 = -2 + 4P_1$$ $$18 + 2 = 4P_1 + 3P_1 - P_2$$ $$20 = 7P_1 - P_2$$ For commodity 2: $$12 + P_1 - 2P_2 = -2 + 3P_2$$ $$12 + 2 + P_1 = 3P_2 + 2P_2$$ $$14 + P_1 = 5P_2$$ 5. **Rewrite the system of equations:** $$7P_1 - P_2 = 20$$ $$P_1 - 5P_2 = -14$$ 6. **Solve the system using substitution or elimination:** Multiply the second equation by 7: $$7P_1 - 35P_2 = -98$$ Subtract the first equation from this: $$(7P_1 - 35P_2) - (7P_1 - P_2) = -98 - 20$$ $$7P_1 - 35P_2 - 7P_1 + P_2 = -118$$ $$-34P_2 = -118$$ 7. **Solve for $P_2$:** $$P_2 = \frac{-118}{-34} = \frac{59}{17}$$ 8. **Substitute $P_2$ back into one equation to find $P_1$:** From $7P_1 - P_2 = 20$: $$7P_1 = 20 + P_2 = 20 + \frac{59}{17} = \frac{340}{17} + \frac{59}{17} = \frac{399}{17}$$ $$P_1 = \frac{399}{17 \times 7} = \frac{399}{119}$$ 9. **Simplify $P_1$ fraction:** $399 = 3 \times 7 \times 19$, $119 = 7 \times 17$, cancel 7: $$P_1 = \frac{3 \times 19}{17} = \frac{57}{17}$$ 10. **Calculate equilibrium quantities:** $$Q_1^* = Q_{s1} = -2 + 4P_1 = -2 + 4 \times \frac{57}{17} = -2 + \frac{228}{17} = \frac{-34}{17} + \frac{228}{17} = \frac{194}{17}$$ $$Q_2^* = Q_{s2} = -2 + 3P_2 = -2 + 3 \times \frac{59}{17} = -2 + \frac{177}{17} = \frac{-34}{17} + \frac{177}{17} = \frac{143}{17}$$ **Final answer:** $$p_1^* = \frac{57}{17}, \quad p_2^* = \frac{59}{17}, \quad Q_1^* = \frac{194}{17}, \quad Q_2^* = \frac{143}{17}$$