At a price of $160 there is demand for 2464 items and a supply of 1440 items.  At a price of $290 there is demand for 1554 items and a supply of 2610 items.  Assuming supply and demand are linear, find the equilibrium price and quantity.

To find the equilibrium price and quantity, we need to set up two linear equations: one for demand and one for supply. The equilibrium occurs when supply equals demand.

### Step 1: Define the variables
Let:
- $ p $ be the price of the item,
- $ q_d $ be the quantity demanded,
- $ q_s $ be the quantity supplied.

### Step 2: Find the demand equation
The demand equation is of the form $ q_d = mp + b $, where $ m $ is the slope and $ b $ is the intercept.

We are given two points on the demand curve:
- At $ p = 160 $, $ q_d = 2464 $,
- At $ p = 290 $, $ q_d = 1554 $.

The slope of the demand curve is calculated as:
\[
m_d = \frac{\Delta q_d}{\Delta p} = \frac{1554 - 2464}{290 - 160} = \frac{-910}{130} = -7.
\]

Thus, the demand equation becomes:
\[
q_d = -7p + b.
\]

Now, substitute $ p = 160 $ and $ q_d = 2464 $ into the equation to solve for $ b $:
\[
2464 = -7(160) + b \implies 2464 = -1120 + b \implies b = 2464 + 1120 = 3584.
\]

So, the demand equation is:
\[
q_d = -7p + 3584.
\]

### Step 3: Find the supply equation
The supply equation is also linear, of the form $ q_s = mp + b $.

We are given two points on the supply curve:
- At $ p = 160 $, $ q_s = 1440 $,
- At $ p = 290 $, $ q_s = 2610 $.

The slope of the supply curve is:
\[
m_s = \frac{\Delta q_s}{\Delta p} = \frac{2610 - 1440}{290 - 160} = \frac{1170}{130} = 9.
\]

Thus, the supply equation becomes:
\[
q_s = 9p + b.
\]

Now, substitute $ p = 160 $ and $ q_s = 1440 $ to solve for $ b $:
\[
1440 = 9(160) + b \implies 1440 = 1440 + b \implies b = 0.
\]

So, the supply equation is:
\[
q_s = 9p.
\]

### Step 4: Find the equilibrium
At equilibrium, supply equals demand, so set $ q_d = q_s $:
\[
-7p + 3584 = 9p.
\]

Solve for $ p $:
\[
3584 = 16p \implies p = \frac{3584}{16} = 224.
\]

Now substitute $ p = 224 $ into either the demand or supply equation to find the equilibrium quantity. Using the supply equation $ q_s = 9p $:
\[
q_s = 9(224) = 2016.
\]

### Final Answer:
The equilibrium price is **$224**, and the equilibrium quantity is **2016 items**.

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Would you like further clarification on any steps?

Here are some related questions you could explore:
1. How would the equilibrium change if demand increases by 20%?
2. What happens to equilibrium if the supply curve has a steeper slope?
3. How do taxes affect the equilibrium price and quantity?
4. Can you graph the supply and demand curves based on the equations derived?
5. What if the demand equation was quadratic—how would you find equilibrium?

**Tip:** In economics, shifts in demand or supply curves will always change the equilibrium price and quantity.

At a price of $160 there is demand for 2464 items and a supply of 1440 items. At a price of $290 there is demand for 1554 items and a supply of 2610 items. Assuming supply and demand are linear, find the equilibrium price and quantity.

Learn how to find the equilibrium price and quantity using linear equations for supply and demand. This problem involves calculating slopes, setting up linear equations, and solving for the intersection point. Perfect for high school students learning algebra and economics.

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