Problem 43 - Contemporary Precalculus

Question

Find the equilibrium quantity and the equilibrium price. In the supply and demand equations, \(p\) is price (in dollars) and \(x\) is quantity (in thousands). Supply: \(p=300-30 x\) Demand: \(p=80+25 x\)

Step-by-Step Solution

Answer

Answer: The equilibrium quantity in this market is 4,000, and the equilibrium price is $180.

Step 1: (Step 1: Set Supply Equation Equal to Demand Equation)

(We begin by equating the supply and demand equations to find the equilibrium point: $$300 - 30x = 80 + 25x$$)

Step 2: (Step 2: Solve for Quantity x)

(Next, we solve the resulting equation for x: $$55x = 220$$ Divide both sides by 55: $$x = 4$$ So the equilibrium quantity is \(4,000\) (since \(x\) represents thousands))

Step 3: (Step 3: Find the Equilibrium Price p)

(Now, we substitute the equilibrium quantity back into either the supply or demand equation to find the equilibrium price. We'll use the supply equation for this example: $$p = 300 - 30(4)$$ $$p = 300 - 120$$ $$p = 180$$ So the equilibrium price is $180.) So, the equilibrium quantity is \(4,000\), and the equilibrium price is $180.

Key Concepts

Supply and Demand Equations

To understand how markets work, it's essential to grasp the basics of supply and demand equations. These equations represent the relationship between the quantity of a good or service that producers are willing to sell at various prices (supply), and the quantity that consumers are willing to buy at those prices (demand).

In the given exercise, the supply equation is defined as p=300-30x, indicating that the price decreases as the quantity increases, which is typical behavior for a supply curve. Conversely, the demand equation p=80+25x illustrates that the price increases with an increase in quantity demanded — a common characteristic of a demand curve. The intercepts and the slope of these linear equations are crucial in determining how much of a product is available at certain price points and how much is desired by the market.

Solving Linear Equations

When you're faced with linear equations in a supply and demand context, the goal is often to find where these two curves intersect, which is the equilibrium point. However, to get there, you'll need to solve the equations for their unknown variables. This process involves isolating the variable of interest on one side of the equation by performing algebraic operations such as adding, subtracting, dividing, or multiplying both sides of the equation by the same amount.

For example, in our exercise, to find the quantity where supply equals demand, you solve the equation 300 - 30x = 80 + 25x for x. This involves combining like terms and simplifying to isolate x. With a good understanding of algebraic principles, solving for x becomes a methodical process that unveils the equilibrium quantity.

Equilibrium Point in Market

The equilibrium point in a market is a powerful concept. It is where the supply of goods matches the demand. When plotted on a graph with price on the y-axis and quantity on the x-axis, the supply and demand curves intersect at the equilibrium point. This point is significant because it reflects a balance in the market where there is no shortage or surplus of goods.

In the provided exercise, after finding that x = 4, we note that this represents the equilibrium quantity (4,000 units due to the 'in thousands' specification). To find the equilibrium price, we plug the value of x into either the supply or demand equation. Here, it's substitution into the supply equation that yields an equilibrium price of $180. Market equilibrium is essential for understanding how supply and demand influence pricing and availability in economics.