Problem 54 - College Algebra and Calculus: An Applied Approach

Question

Supply and Demand The supply and demand equations for a microscope are given by \(\left\\{\begin{array}{ll}p+0.85 x=650 & \text { Demand } \\ p-0.4 x=75 & \text { Supply }\end{array}\right.\) where \(p\) is the price (in dollars) and \(x\) represents the number of microscopes. For how many units will the quantity demanded equal the quantity supplied? What price corresponds to this value?

Step-by-Step Solution

Answer

The quantity at which the supply equals the demand is approximately 460 microscopes. The price corresponding to this value is approximately $259.

Step 1: Set up the system of equations

Initially, there are the two initial equations: \( p + 0.85x = 650 \) (Demand) \( p - 0.4x = 75 \) (Supply) The task is to find the values \( p \) (the price when the supply equals the demand) and \( x \) (the number of microscopes at this price). For this, the two equations need to be simultaneously solved.

Step 2: Subtract the Supply equation from the Demand equation

By subtracting the Supply equation from the Demand equation, it helps isolate \( x \). \( (p + 0.85x) - (p - 0.4x) = (650 - 75) The \( p \) in both equations cancels out, leaving: \( 0.85x + 0.4x = 575 Which simplifies to: \( 1.25x = 575 \)

Step 3: Solve for \( x \)

Now solve for \( x \) by dividing both sides with 1.25: \( x = \frac{575}{1.25} After performing the division, find that \( x \approx 460 \). This is the number of microscopes for which the supply equals the demand.

Step 4: Substitute \( x \) into the supply equation and solve for \( p \)

Finally, substitute \( x \) into one of the original equations, for instance, the supply equation and then solve for \( p \). This gives: \( p - 0.4*460 = 75 Or: \( p = 75 + 0.4*460 Compute this and find that \( p \) is approximately 259. This is the price in dollars at which the supply equals the demand.

Key Concepts

System of Equations

The system of equations is at the heart of solving supply and demand problems. In this case, it consists of two equations representing the demand and supply:

• Demand: \( p + 0.85x = 650 \)
• Supply: \( p - 0.4x = 75 \)

Each equation represents a relationship between the price \( p \) and the quantity \( x \), showing how they interact with each other.

To find the economic equilibrium of a market, it is crucial to solve these equations simultaneously. That means finding a pair \( (p, x) \) that satisfies both equations at the same time. This indicates the point where the quantity demanded equals the quantity supplied.

Price and Quantity

In economics, price and quantity are interconnected. The demand equation shows us how price is influenced by quantity, while the supply equation shows how quantity affects price.

In our system:

• The variable \( p \) represents the price of the microscopes in dollars.
• The variable \( x \) represents the number of microscopes involved.

Understanding this relationship helps in knowing how changes in one can affect the other and forms the basis for many economic models. By determining the values of \( p \) and \( x \), we can find the equilibrium condition where both the supply and demand are in harmony.

Algebraic Solution

Solving the system algebraically involves manipulating the equations to isolate one variable. For our supply and demand problem, this means:

• First, subtract the supply equation from the demand equation to eliminate the price \( p \).
• This gives a new equation with only \( x \): \( 0.85x + 0.4x = 575 \), which simplifies to \( 1.25x = 575 \).
• Next, solve for \( x \) by dividing both sides by 1.25, to find the quantity \( x = 460 \).

Once \( x \) is found, substitute it back into either original equation to solve for \( p \). In this case, substituting into the supply equation simplifies the process, leading to \( p = 259 \).
This algebraic approach ensures a systematic way to find the solution where the supply and demand are equal.

Economic Equilibrium

Economic equilibrium occurs when supply matches demand, both in quantity and price. This is the point where the interests of buyers and sellers align, creating a stable market state. In our microscope example:

• We found the equilibrium quantity to be \( x = 460 \), meaning 460 microscopes are traded.
• The equilibrium price is \( p = 259 \), representing the market price where this quantity is both demanded and supplied.

Achieving this balance is crucial in real markets to prevent surplus or shortage. Understanding equilibrium helps in making predictions about how changes in external factors might affect a market. The algebraic solution not only gives a specific equilibrium point but also illustrates the delicate balance between supply and demand.