At a price of $27.75, producers will provide 8 items, while at a price of $33.25, they will provide 30 items.
Consumers will purchase 85 of these items if the price is $91.5, but will purchase 133 items if the price decreases to $81.25.
Find the market equilibrium point. (Enter your answers as a comma-separated list.)


The market equilibrium point occurs where the quantity supplied equals the quantity demanded. To find this point, we first need to determine the supply and demand equations.

### Step 1: Supply Equation
The supply equation is typically linear and can be written as:
\[
Q_s = m_s P + b_s
\]
where $ Q_s $ is the quantity supplied, $ P $ is the price, $ m_s $ is the slope, and $ b_s $ is the y-intercept.

Using the two points provided for supply:
- At $ P = 27.75 $, $ Q_s = 8 $
- At $ P = 33.25 $, $ Q_s = 30 $

We can find the slope $ m_s $ by using the slope formula:
\[
m_s = \frac{\Delta Q_s}{\Delta P} = \frac{30 - 8}{33.25 - 27.75} = \frac{22}{5.5} = 4
\]
Thus, $ m_s = 4 $.

Now substitute $ m_s = 4 $ and one of the points (let's use $ P = 27.75 $ and $ Q_s = 8 $) into the supply equation to find $ b_s $:
\[
8 = 4(27.75) + b_s
\]
\[
8 = 111 + b_s
\]
\[
b_s = 8 - 111 = -103
\]
So, the supply equation is:
\[
Q_s = 4P - 103
\]

### Step 2: Demand Equation
The demand equation is also linear and can be written as:
\[
Q_d = m_d P + b_d
\]
where $ Q_d $ is the quantity demanded, and $ m_d $ and $ b_d $ are the slope and intercept for demand.

Using the two points provided for demand:
- At $ P = 91.5 $, $ Q_d = 85 $
- At $ P = 81.25 $, $ Q_d = 133 $

We find the slope $ m_d $ using the slope formula:
\[
m_d = \frac{\Delta Q_d}{\Delta P} = \frac{133 - 85}{81.25 - 91.5} = \frac{48}{-10.25} \approx -4.683
\]
Thus, $ m_d \approx -4.683 $.

Now substitute $ m_d = -4.683 $ and one of the points (let's use $ P = 91.5 $ and $ Q_d = 85 $) into the demand equation to find $ b_d $:
\[
85 = -4.683(91.5) + b_d
\]
\[
85 = -428.47 + b_d
\]
\[
b_d = 85 + 428.47 = 513.47
\]
So, the demand equation is:
\[
Q_d = -4.683P + 513.47
\]

### Step 3: Find the Equilibrium Point
To find the equilibrium, set $ Q_s = Q_d $:
\[
4P - 103 = -4.683P + 513.47
\]
Solve for $ P $:
\[
4P + 4.683P = 513.47 + 103
\]
\[
8.683P = 616.47
\]
\[
P = \frac{616.47}{8.683} \approx 71.00
\]

Now substitute $ P = 71.00 $ into either the supply or demand equation to find $ Q $. Using the supply equation:
\[
Q_s = 4(71.00) - 103 = 284 - 103 = 181
\]

Thus, the equilibrium price is $ 71.00 $ and the equilibrium quantity is $ 181 $.

### Final Answer:
The market equilibrium point is $ (71.00, 181) $.

Would you like further details or have any questions?

Here are 5 related questions:
1. How do changes in supply or demand affect the equilibrium point?
2. What happens if the supply equation has a different slope?
3. Can a non-linear supply or demand curve lead to multiple equilibrium points?
4. How does government intervention (e.g., price ceiling or floor) affect equilibrium?
5. What is the impact of taxes on the equilibrium price and quantity?

**Tip:** Always check that supply equals demand at the calculated equilibrium to ensure accuracy.

At a price of $27.75, producers will provide 8 items, while at a price of $33.25, they will provide 30 items. Consumers will purchase 85 of these items if the price is $91.5, but will purchase 133 items if the price decreases to $81.25. Find the market equilibrium point. (Enter your answers as a comma-separated list.)

Discover how to find the market equilibrium where supply equals demand using linear equations. This problem involves determining the supply and demand curves from given data points and solving for the equilibrium price and quantity. Suitable for students in Grades 10-12.

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