1. Let's start with a common problem in mathematical economics: finding the equilibrium price in a market where demand and supply are given by linear functions. 2. Suppose the demand function is $Q_d = 100 - 2P$ and the supply function is $Q_s = 3P - 20$, where $Q$ is quantity and $P$ is price. 3. The equilibrium price occurs where quantity demanded equals quantity supplied, so we set $Q_d = Q_s$: $$100 - 2P = 3P - 20$$ 4. To solve for $P$, add $2P$ to both sides and add $20$ to both sides: $$100 + 20 = 3P + 2P$$ $$120 = 5P$$ 5. Divide both sides by 5: $$P = \frac{120}{5} = 24$$ 6. The equilibrium price is $24$. 7. To find the equilibrium quantity, substitute $P=24$ into either the demand or supply function: $$Q_d = 100 - 2(24) = 100 - 48 = 52$$ 8. So, the equilibrium quantity is $52$ units. This example shows how to find equilibrium in a market using algebraic methods.