Put demand and supply on one graph. Where they cross, buyers' and sellers' plans match. Finding that point with equations is the skill from Variables on Both Sides: set the two quantities equal. You know supply from The Law of Supply.
Key idea
The equilibrium price $P^*$ is the price at which quantity demanded equals quantity supplied; that common amount is the equilibrium quantity $Q^*$. For $Q_d = a - bP$ and $Q_s = c + dP$: $$a - bP = c + dP \;\Rightarrow\; P^* = \frac{a - c}{b + d}, \qquad Q^* = a - bP^*.$$
Why it works
At any other price, the plans don't match, and someone has a reason to change the price. Above $P^*$, sellers offer more than buyers want, unsold goods pile up and sellers cut prices. Below $P^*$, buyers want more than is offered, they bid prices up, and sellers raise them. Only at $P^*$ does nobody have that reason. That is what "equilibrium" means: a resting point.
Worked example: Solve for the equilibrium
$Q_d = 20 - 2P$ and $Q_s = -4 + 2P$. Find the equilibrium price and quantity.
- Set them equal: $20 - 2P = -4 + 2P$.
- Add $2P$ and 4 to both sides: $24 = 4P$, so $P^* = 6$.
- Quantity: $Q_d = 20 - 2(6) = 8$. Check with supply: $-4 + 2(6) = 8$. ✓
Answer: $P^* = \$6$, $Q^* = 8$.
Worked example: Inverse forms
Demand is $P = 15 - Q$ and supply is $P = 3 + 0.5Q$. Find the equilibrium.
- Here each equation gives a price, so set the prices equal: $15 - Q = 3 + 0.5Q$.
- $12 = 1.5Q$, so $Q^* = 8$.
- $P^* = 15 - 8 = 7$. Check: $3 + 0.5(8) = 7$. ✓
Answer: $Q^* = 8$, $P^* = \$7$.
Equilibrium doesn't mean everyone is happy with the price; it means no one has a reason to change it. Real markets take time to get there, and they keep moving because the curves keep shifting, which is the subject of the next lessons.
Common mistake
Solving $Q_d = Q_s$ for the price and then reporting that number as the quantity, or plugging the price into only one equation. In the first example, $P = 6$ but $Q = 8$. Check: put $P^*$ into both equations; they must give the same quantity, and that quantity is $Q^*$.