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

Q = D(p) = 1088−13p
, 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.

To analyze the market equilibrium for widgets, we need to determine the price \( p \) at which the quantity demanded equals the quantity supplied. This is where the demand and supply curves intersect, meaning \( D(p) = S(p) \).

### 1. Setting the demand equal to the supply:

\[
D(p) = S(p)
\]

Given:
\[
1088 - 13p = -4 + 8p
\]

### 2. Solving for \( p \):

First, we add \( 13p \) to both sides to collect the \( p \) terms on one side:
\[
1088 = -4 + 21p
\]

Next, we add 4 to both sides:
\[
1092 = 21p
\]

Finally, divide by 21 to solve for \( p \):
\[
p = \frac{1092}{21} \approx 52
\]

### 3. Determining the equilibrium quantity:

Now that we have the equilibrium price \( p \), substitute it back into either the demand or supply function to find the equilibrium quantity \( Q \).

Using the supply function \( S(p) \):
\[
Q = S(52) = -4 + 8(52) = -4 + 416 = 412
\]

### Summary:

- **Equilibrium Price**: \( p = 52 \)
- **Equilibrium Quantity**: \( Q = 412 \)

This means that at a price of 52, the market will demand and supply 412 widgets.

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Would you like to see more details on any part of this solution or ask any specific questions?

Here are 5 related questions you might find useful:
1. What happens to the equilibrium price and quantity if the demand curve shifts?
2. How would a price ceiling below the equilibrium price affect the market?
3. What is the consumer surplus at the equilibrium price?
4. How does elasticity of demand or supply impact the equilibrium?
5. Can you analyze the effects of a subsidy on the supply curve?

**Tip:** Understanding how shifts in demand or supply curves affect equilibrium helps in analyzing real-world market scenarios.

Assume that the demand curve D(p) given below is the market demand for widgets: Q = D(p) = 1088−13p, 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.

Discover how to determine the market equilibrium for widgets by solving the equations for demand and supply functions. This problem involves algebraic concepts, specifically linear equations, to find the equilibrium price and quantity. Ideal for students in Grades 11-12.

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