Assume that the demand curve D(p) given below is the market demand for widgets:

Q=D(p)=1496−12p𝑄=𝐷(𝑝)=1496-12𝑝, p > 0

Let the market supply of widgets be given by:

Q=S(p)=−4+8p𝑄=𝑆(𝑝)=-4+8𝑝, p > 0

where p is the price and Q is the quantity. The functions D(p) and S(p) give the number of widgets demanded and supplied at a given price.

To analyze this situation, we have the following:

- **Demand function**: \( D(p) = 1496 - 12p \)
- **Supply function**: \( S(p) = -4 + 8p \)

Where:
- \( p \) is the price.
- \( Q \) is the quantity of widgets.

### 1. **Market Equilibrium**

To find the equilibrium price \( p_e \) and quantity \( Q_e \), we set the demand equal to the supply, i.e., \( D(p) = S(p) \), since at equilibrium, the quantity demanded equals the quantity supplied.

\[
1496 - 12p = -4 + 8p
\]

Now, let's solve for \( p_e \).

\[
1496 + 4 = 12p + 8p
\]

\[
1500 = 20p
\]

\[
p_e = \frac{1500}{20} = 75
\]

So, the equilibrium price is \( p_e = 75 \).

### 2. **Equilibrium Quantity**

Now substitute \( p_e = 75 \) into either the demand or supply function to find the equilibrium quantity \( Q_e \).

Using the demand function:
\[
Q_e = D(75) = 1496 - 12(75)
\]
\[
Q_e = 1496 - 900 = 596
\]

Thus, the equilibrium quantity is \( Q_e = 596 \).

### Summary:
- **Equilibrium price**: \( p_e = 75 \)
- **Equilibrium quantity**: \( Q_e = 596 \)

Would you like more details on any part of this explanation, or any specific topic expanded?

### 5 Related Questions:
1. How does a shift in the demand curve affect the equilibrium price and quantity?
2. What happens if the government imposes a price ceiling below the equilibrium price?
3. How would the equilibrium change if the supply function were steeper (more inelastic)?
4. What does the slope of the demand function tell us about the price elasticity of demand?
5. How does the presence of taxes or subsidies affect market equilibrium?

### Tip:
To understand how shifts in demand and supply affect market equilibrium, consider how the functions change relative to price—this gives insights into elasticity and market reactions.

Assume that the demand curve D(p) given below is the market demand for widgets: Q=D(p)=1496−12p, p > 0. Let the market supply of widgets be given by: Q=S(p)=-4+8p, p > 0, where p is the price and Q is the quantity. The functions D(p) and S(p) give the number of widgets demanded and supplied at a given price.

Solve for the market equilibrium price and quantity for widgets using the demand function D(p) = 1496−12p and the supply function S(p) = -4+8p. This problem covers core algebraic concepts such as solving linear equations and understanding market equilibrium. Suitable for students in Grades 10-12.

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