Problem 108 - A Graphical Approach to Precalculus with Limits

Question

Find the equilibrium price in dollars if$$p=\frac{2}{3} q \quad \text { and } \quad p=49-\frac{1}{2} q$$ How many units represent the demand at this price?

Step-by-Step Solution

Answer

The equilibrium price is $28, and the demand is 42 units.

Step 1: Set the Equations Equal

The equilibrium price occurs when the supply and demand equal each other. To find this, set the two price equations equal to each other: \[\frac{2}{3}q = 49 - \frac{1}{2}q\]

Step 2: Eliminate the Fraction

To eliminate fractions, multiply every term by 6, which is the least common multiple of the denominators: \[6 \times \frac{2}{3}q = 6 \times 49 - 6 \times \frac{1}{2}q\] This simplifies to: \[4q = 294 - 3q\]

Step 3: Solve for q

Rearrange the equation to solve for \(q\): \[4q + 3q = 294\] \[7q = 294\] Divide both sides by 7: \[q = 42\]

Step 4: Substitute q to Find the Equilibrium Price

Substitute \(q = 42\) back into either original equation to find \(p\). Using \(p = \frac{2}{3} q\): \[p = \frac{2}{3} \times 42\] \[p = 28\]

Step 5: Verify the Equilibrium

Verify that substituting \(q = 42\) into \(p = 49 - \frac{1}{2}q\) also gives \(p = 28\): \[p = 49 - \frac{1}{2} \times 42\] \[p = 49 - 21 = 28\] Both methods confirm that the equilibrium price is correct.

Key Concepts

Supply and Demand

In economics, supply and demand describe the relationship between the quantity of a product available and the desire of consumers to buy. This relationship determines the market price. At the core, the **law of supply and demand** suggests that they reach an equilibrium at a certain price point, known as the **equilibrium price**.

When a market reaches its equilibrium price, the quantity of goods supplied equals the quantity demanded. This balance ensures there is neither excess supply nor excess demand. Understanding this concept helps businesses and consumers make informed decisions.

The supply and demand curves represent these concepts graphically.

• The **supply curve** usually slopes upwards, indicating higher prices motivate producers to supply more.
• The **demand curve** typically slopes downwards, as higher prices deter consumers from purchasing.

The intersection of the supply and demand curves is where the equilibrium price is found. In this exercise, the provided linear equations depict such relationships, allowing us to calculate the point of intersection, representing the equilibrium price.

Linear Equations

Linear equations are mathematical expressions that depict a straight line when graphed. They are often used in economics to represent relationships such as supply and demand.

These equations can be written in the form of \(y = mx + b\), where \(m\) is the slope, and \(b\) is the y-intercept. However, in our context, the variables are often replaced with meaningful economic factors, like price \(p\) and quantity \(q\).

Let's break down the two linear equations from the exercise:

• **Equation 1:** \(p = \frac{2}{3}q\) - This reflects a direct relationship where the price \(p\) increases with quantity \(q\).
• **Equation 2:** \(p = 49 - \frac{1}{2}q\) - Here, the price \(p\) decreases as the quantity \(q\) increases.

Solving these linear equations helps find how the equilibrium is reached. Step 1 involves setting the equations equal to each other, as this identifies where both supply and demand meet at the same price and quantity. This is critical in finding the equilibrium price.

Fractions

Fractions represent a part of a whole and are crucial in many areas of mathematics, including solving linear equations in economics. They can initially seem complex, but they provide precise values that decimals sometimes cannot.

In this exercise, fractions appear in both equations, such as \(\frac{2}{3}\) and \(\frac{1}{2}\). To simplify problem-solving, we often eliminate fractions by finding a common denominator or multiplying through by an appropriate number, as seen in Step 2 of the solution.

• The **common denominator** allows us to clear fractions to work with whole numbers, simplifying arithmetic operations.
• By multiplying each term by the least common multiple, we transformed:\[6 \times \frac{2}{3}q = 6 \times 49 - 6 \times \frac{1}{2}q\]into an equation without fractions.

This step is crucial as it simplifies equation manipulation, leading to easier calculation of the equilibrium price and quantity. Mastering fractions helps students tackle similar tasks across different mathematical and real-world contexts with confidence.