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find equillbum then do find supply and demand equations

Let's work through the problem step by step.

### (a) Identify Equilibrium Price and Quantity
The equilibrium price and quantity occur where the supply and demand curves intersect on the graph.

- **Equilibrium Price**: The price at which the supply and demand curves intersect.
- **Equilibrium Quantity**: The quantity at which the supply and demand curves intersect.

From the graph:
- **Equilibrium Price** = \$90
- **Equilibrium Quantity** = 20 backpacks

### (b) Derive the Demand Equation
The demand curve is typically downward sloping. Let's derive the equation in the form $ P = mQ + b $, where $ P $ is the price, $ Q $ is the quantity demanded, $ m $ is the slope, and $ b $ is the y-intercept.

To find the slope ($ m $):
\[ m = \frac{\Delta P}{\Delta Q} = \frac{P_2 - P_1}{Q_2 - Q_1} \]

Using two points on the demand curve:
- Point 1: $ (Q_1, P_1) = (0, 150) $
- Point 2: $ (Q_2, P_2) = (40, 0) $

So,
\[ m = \frac{0 - 150}{40 - 0} = \frac{-150}{40} = -\frac{15}{4} \]

The demand equation is therefore:
\[ P = -\frac{15}{4}Q + 150 \]

### (c) Derive the Supply Equation
The supply curve is typically upward sloping. The supply equation is also of the form $ P = mQ + b $.

Using two points on the supply curve:
- Point 1: $ (Q_1, P_1) = (0, 30) $
- Point 2: $ (Q_2, P_2) = (40, 150) $

So,
\[ m = \frac{150 - 30}{40 - 0} = \frac{120}{40} = 3 \]

The supply equation is therefore:
\[ P = 3Q + 30 \]

### (d) Numerically Solve for Equilibrium
To find the equilibrium algebraically, set the supply equation equal to the demand equation:

\[ -\frac{15}{4}Q + 150 = 3Q + 30 \]

Solve for $ Q $:

\[ 150 - 30 = 3Q + \frac{15}{4}Q \]

\[ 120 = 3Q + \frac{15}{4}Q \]

To eliminate the fraction, multiply the entire equation by 4:

\[ 480 = 12Q + 15Q \]

\[ 480 = 27Q \]

\[ Q = \frac{480}{27} \approx 17.78 \text{ (approximately 18 backpacks)} \]

Substitute $ Q $ back into either the demand or supply equation to find $ P $:

Using the demand equation:
\[ P = -\frac{15}{4}(17.78) + 150 \approx 90 \text{ (approximately)} \]

This result is consistent with the graph.

### Would you like more details or have any questions about this?

Here are five related questions you might consider:
1. What would happen to the equilibrium if the supply curve shifted to the right?
2. How does the slope of the demand curve affect the elasticity of demand?
3. What would be the new equilibrium price if the demand increased to $ P = -\frac{15}{4}Q + 160 $?
4. How would taxes on producers affect the supply curve and equilibrium?
5. Can you explain how price ceilings might affect the market shown?

**Tip:** Understanding how to derive supply and demand equations can help in analyzing shifts in the curves and predicting market outcomes under different conditions.

Learn how to find equilibrium price and quantity, derive demand and supply equations, and solve for equilibrium algebraically with step-by-step guidance. This problem involves understanding the intersection of supply and demand curves, deriving linear equations for demand and supply, and solving linear equations algebraically. Suitable for advanced high school students and beyond.

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