1. **Problem Statement:** Find the equilibrium price $P$ and quantity $Q$ given demand $Q_d = 50 - 2P$ and supply $Q_s = -10 + 3P$. 2. **Formula and Rules:** At equilibrium, quantity demanded equals quantity supplied: $$Q_d = Q_s$$ 3. **Set equations equal:** $$50 - 2P = -10 + 3P$$ 4. **Solve for $P$:** Add $2P$ to both sides: $$50 = -10 + 5P$$ Add $10$ to both sides: $$60 = 5P$$ Divide both sides by $5$: $$P = \frac{60}{5} = 12$$ 5. **Find equilibrium quantity $Q$:** Substitute $P=12$ into either demand or supply equation: $$Q_d = 50 - 2(12) = 50 - 24 = 26$$ 6. **Answer:** The equilibrium price is $12$ and the equilibrium quantity is $26$ units.