a price of $26.75, producers will provide 9 items, while at a price of $32.25, they will provide 31 items. Consumers will purchase 86 of these items if the price is $89.5, but will purchase 132 items if the price decreases to $79.25. Find the market equilibrium point. (Enter your answers as a comma-separated list.)

The market equilibrium point is found where the supply and demand functions intersect, meaning the quantity supplied equals the quantity demanded at a specific price.

### Step 1: Find the supply function

The supply function typically has the form:
\[
q_s = m_s p + b_s
\]
Where:
- $ q_s $ is the quantity supplied,
- $ p $ is the price,
- $ m_s $ is the slope of the supply line,
- $ b_s $ is the intercept (quantity when price is zero).

We are given two points on the supply curve:
\[
(26.75, 9) \quad \text{and} \quad (32.25, 31)
\]

1. The slope $ m_s $ of the supply function is calculated using the formula:
\[
m_s = \frac{\Delta q_s}{\Delta p} = \frac{31 - 9}{32.25 - 26.75} = \frac{22}{5.5} = 4
\]
2. To find the intercept $ b_s $, substitute one of the points into the supply function $ q_s = m_s p + b_s $. Using the point $ (26.75, 9) $:
\[
9 = 4(26.75) + b_s
\]
\[
9 = 107 + b_s
\]
\[
b_s = 9 - 107 = -98
\]
Thus, the supply function is:
\[
q_s = 4p - 98
\]

### Step 2: Find the demand function

The demand function has the form:
\[
q_d = m_d p + b_d
\]
Where:
- $ q_d $ is the quantity demanded,
- $ p $ is the price,
- $ m_d $ is the slope of the demand line,
- $ b_d $ is the intercept.

We are given two points on the demand curve:
\[
(89.5, 86) \quad \text{and} \quad (79.25, 132)
\]

1. The slope $ m_d $ of the demand function is:
\[
m_d = \frac{\Delta q_d}{\Delta p} = \frac{132 - 86}{79.25 - 89.5} = \frac{46}{-10.25} \approx -4.49
\]
2. To find the intercept $ b_d $, substitute one of the points into the demand function $ q_d = m_d p + b_d $. Using the point $ (89.5, 86) $:
\[
86 = -4.49(89.5) + b_d
\]
\[
86 = -402.355 + b_d
\]
\[
b_d = 86 + 402.355 \approx 488.36
\]
Thus, the demand function is:
\[
q_d = -4.49p + 488.36
\]

### Step 3: Find the equilibrium point

To find the equilibrium, set the supply equal to the demand:
\[
4p - 98 = -4.49p + 488.36
\]
Solve for $ p $:
\[
4p + 4.49p = 488.36 + 98
\]
\[
8.49p = 586.36
\]
\[
p \approx \frac{586.36}{8.49} \approx 69.06
\]

Now, substitute $ p = 69.06 $ into the supply function to find the quantity:
\[
q_s = 4(69.06) - 98 \approx 276.24 - 98 = 178.24
\]

Thus, the equilibrium price is approximately $ 69.06 $, and the equilibrium quantity is approximately $ 178.24 $.

### Final Answer:
The market equilibrium point is approximately $ (69.06, 178.24) $.

Would you like more details or have any questions?

Here are five related questions to explore:
1. How does changing the slope of the demand curve affect the equilibrium point?
2. What happens to the equilibrium price if the supply curve becomes steeper?
3. How can government interventions, such as price floors, impact market equilibrium?
4. How would a simultaneous shift in both supply and demand curves affect the equilibrium?
5. What are some real-world examples of markets that quickly reach equilibrium?

**Tip:** In a competitive market, any excess supply or demand will adjust over time to restore equilibrium.

A price of $26.75, producers will provide 9 items, while at a price of $32.25, they will provide 31 items. Consumers will purchase 86 of these items if the price is $89.5, but will purchase 132 items if the price decreases to $79.25. Find the market equilibrium point. (Enter your answers as a comma-separated list.)

Learn how to find the market equilibrium point where supply and demand intersect. In this example, producers supply 9 items at $26.75 and 31 items at $32.25. Consumers demand 86 items at $89.5 and 132 items at $79.25. Follow the step-by-step process to calculate the equilibrium price and quantity.

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