Problem 76 - College Algebra

Question

The law of supply and demand states that, in a free market economy, a commodity tends to be sold at its equilibrium price. At this price, the amount that the seller will supply is the same amount that the consumer will buy. Explain how systems of equations can be used to determine the equilibrium price.

Step-by-Step Solution

Answer

Systems of equations are used to determine the equilibrium price by setting the supply and demand equations equal to each other, representing the point where demand equals supply. Solving this system yields the equilibrium price and quantity.

Step 1: Understand the Problem

This is a conceptual problem asking about the application of systems of equations in the realm of economics. Specifically, it inquires about the use of these mathematical tools in finding an equilibrium price where demand and supply are equal.

Step 2: Define the Variables

Normally, price (P) and quantity (Q) are the variables used in supply and demand equations in economics. Usually these models are linear and take the form \( Q = mP + c \), where \( m \) and \( c \) are the slope and intercept of the line respectively.

Step 3: Set up the Equations

Once the demand and supply equations have been established based on empirical data or theoretical considerations, the next step is to set them equal to each other. This gives a system of equations that can be solved simultaneously. In the context of supply and demand, this represents the quantity supplied being equal to the quantity demanded, which is the definition of an equilibrium state in a market.

Step 4: Solve the Equations

The solution to the system of equations will give the equilibrium price (P) and the quantity (Q). Various methods for solving systems of equations exist including substitution, elimination, and graphing.

Step 5: Interpret the Result

The equilibrium price is the price at which the quantity demanded by buyers equals the quantity that sellers are willing to offer. At this price, total surplus (the sum of consumer surplus and producer surplus) in the market is maximized. Additionally, there is no excess supply or demand in the market at the equilibrium price.