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Let's start by solving the given system of supply and demand equations:

### Supply equation:
\[
y = 2x + 506
\]

### Demand equation:
\[
y = -3x + 641
\]

#### Step 1: Find the intersection point (equilibrium)
At the equilibrium, supply equals demand, so we can set the two equations equal to each other:

\[
2x + 506 = -3x + 641
\]

Now, solve for $ x $.

1. Add $ 3x $ to both sides:
\[
2x + 3x + 506 = 641
\]
\[
5x + 506 = 641
\]

2. Subtract 506 from both sides:
\[
5x = 641 - 506
\]
\[
5x = 135
\]

3. Divide both sides by 5:
\[
x = \frac{135}{5} = 27
\]

So, $ x = 27 $.

#### Step 2: Find the corresponding $ y $-value
Now that we know $ x = 27 $, substitute this value into either the supply or demand equation to find $ y $. Let's use the supply equation:

\[
y = 2(27) + 506
\]
\[
y = 54 + 506 = 560
\]

So, the equilibrium point is $ (27, 560) $.

#### Step 3: Answer the questions

- **The intersection point (equilibrium)** is $ (27, 560) $.
- **The selling price** when supply and demand are in equilibrium is $ \$27 $ per item (this is the value of $ x $).
- **The amount of items in the market** when supply and demand are in equilibrium is 560 items (this is the value of $ y $).

---

### Recap of answers:
1. **Intersection point**: $ (27, 560) $
2. **Selling price**: $ \$27 $ per item
3. **Number of items**: 560

---

Do you want more details or have any questions?

Here are 5 related questions:
1. How would a change in supply affect the equilibrium price?
2. What happens if the demand equation changes to $ y = -4x + 700 $?
3. How can we model a shift in the demand curve using this system of equations?
4. If the supply equation changes, how do we find the new equilibrium?
5. What is the interpretation of the slope in the supply and demand equations?

**Tip**: When solving systems of linear equations, it's often helpful to first isolate one variable, then substitute back to find the second.

What is the intersection point of the supply and demand equations, and what are the equilibrium price and quantity?

Discover how to solve for the intersection point and equilibrium in supply and demand equations. This problem involves finding the equilibrium price and quantity where supply meets demand, suitable for high school students learning algebra and economics.

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