Solve equilibrium equations without losing the economics

The algebra is only a compact version of the graph.

Suppose demand is Q_d=100-5P and supply is Q_s=20+3P. Equilibrium requires the same quantity to be both purchased and sold, so set the expressions equal: 100-5P=20+3P. Solving gives P=10, and substitution into either equation gives Q=50. Always check both equations. A price that produces different planned quantities is not an equilibrium, even if the arithmetic looks neat.

Keep the economic roles visible while rearranging: 100-5P=20+3P 80=8P P_e=10. Only then substitute for quantity. Adding the slopes or intercepts without first writing equality often produces a number with no clear interpretation. The equality is the algebraic version of the curve intersection.

Now suppose an increase in buyer income changes demand to Q_d=116-5P while supply is unchanged. The new solution is P=12 and Q=56. The numbers confirm the graph’s prediction: demand shifted right, so both equilibrium price and quantity rose. Algebra should reinforce the mechanism, not replace it.

A per-unit tax can be represented in equations by separating the buyer price from the seller price. If the tax is t, then P_b-P_s=t. Demand uses P_b and supply uses P_s. Solving Q_d(P_b)=Q_s(P_s) together with the wedge equation prevents the common mistake of inserting the same price into both sides after a tax.

An equation can also identify imbalance at a trial price. At P=8, the original demand equation gives 60 and supply gives 44, so the shortage is 16. The market pressure is upward because buyers’ plans exceed sellers’ plans. At P=12, demand is 40 and supply is 56, so the surplus is 16 and pressure is downward.

A supply shift in equation form

Original demand is Q_d=90-3P and supply is Q_s=10+2P, giving P=16 and Q=42. A productivity improvement changes supply to Q_s=20+2P. The new equilibrium solves 90-3P=20+2P, so P=14 and Q=48. Lower price and higher quantity confirm a rightward supply shift.

For absolute beginners, the safest routine is to write one label beside every number: P_e, Q_e, Q_d, or Q_s. That small discipline prevents the most common mistakes-reporting a planned quantity as the equilibrium price, adding quantities instead of subtracting them, or finding the new equilibrium without comparing it with the old one.

Check reasonableness before accepting the answer. With downward-sloping demand and upward-sloping supply, a rightward demand shift should not lower both price and quantity. At a positive equilibrium price, substitute back into both equations and confirm a nonnegative common quantity. If one equation gives 50 and the other 42, the price is not the intersection regardless of how clean the arithmetic appears.

Equations may use inverse form, such as P=20-0.2Q and P=4+0.1Q. Set the two price expressions equal because the equilibrium has one market price. Solve for Q first, then substitute for P. The rule remains “same price and same quantity” even when the algebra presents the curves differently.

Do not skip the initial equilibrium

Mark the old intersection before shifting a curve. Without a starting point, students often compare the new curve with an axis rather than compare the new market outcome with the old one.