Linear Equations and Market Equilibrium

A linear equation in one variable expresses a straight-line relationship of the form $y=mx+c$, where $m$ is the slope (rate of change of y per unit change in x) and $c$ is the intercept (the value of y when x=0). Many core economic relationships — demand, supply, consumption functions — are modelled as linear equations precisely because they are the simplest functional form that still captures a genuine cause-and-effect direction (positive or negative) between two variables.

When two linear equations in the SAME two variables must both hold simultaneously, they are called simultaneous equations, and solving them means finding the one pair of values that satisfies both equations at once. The classic economic application is finding market equilibrium: given a demand function $Q_d=a-bP$ and a supply function $Q_s=c+dP$, the equilibrium price and quantity are found by setting $Q_d=Q_s$ (the condition that the amount buyers wish to buy exactly equals the amount sellers wish to sell) and solving the resulting single equation in P, then substituting back to find Q.

Method (substitution):

1. Set the two expressions equal: $a-bP=c+dP$.
2. Collect all P-terms on one side: $a-c=dP+bP=(b+d)P$.
3. Solve for the equilibrium price: $P^{*}=\dfrac{a-c}{b+d}$.
4. Substitute $P^{*}$ back into EITHER original equation to find the equilibrium quantity $Q^{*}$. …