https://file.aiquickdraw.com/mathbot/file/17299931620551s76aq1o.jpg

To solve this problem, let’s define variables and set up a system of equations based on the information given.

### Problem Analysis
1. **Selling Price per Cake**: $12.
2. **Fixed Cost (Booth Cost)**: $50.
3. **Variable Cost (Ingredients per Cake)**: $2.

We’re asked to find the point where the club's **sales** equal their **expenditures**, meaning **revenue** equals **total cost**.

### Define Variables
- Let $ x $ represent the number of cakes sold.
- Let $ y $ represent the amount in dollars.

### Equations
1. **Revenue Equation**: The revenue from selling $ x $ cakes at $12 each is:
   \[
   y = 12x
   \]

2. **Cost Equation**: The total cost includes the fixed booth cost ($50) plus $2 for each cake, giving:
   \[
   y = 2x + 50
   \]

### Solution: Finding the Break-even Point
To find where revenue equals costs, set the two equations equal to each other:
\[
12x = 2x + 50
\]

Solving for $ x $:
\[
12x - 2x = 50
\]
\[
10x = 50
\]
\[
x = 5
\]

Now, substitute $ x = 5 $ into either equation to find $ y $:
\[
y = 12 \times 5 = 60
\]

### Conclusion
The club members need to sell **5 cakes** to cover their expenditures, and both the sales and expenditures will be **$60** at this point.

### Graphing
1. **Revenue Line**: Plot the line $ y = 12x $.
2. **Cost Line**: Plot the line $ y = 2x + 50 $.
3. The intersection point at $ (5, 60) $ represents the break-even point.

Would you like more details on the graph or calculations?

### Related Questions
1. How would the break-even point change if the cost per cake increased?
2. What if the booth fee was reduced to $30? How many cakes would they need to sell then?
3. If the selling price per cake decreased, how would that affect the break-even point?
4. How does this model change if the booth fee is waived entirely?
5. Can we calculate profit if they sell more than 5 cakes?

#### Tip:
In cost-revenue analysis, always separate fixed and variable costs for a clear understanding of break-even points.

The members of a cooking club are making cakes, which they will sell at a street fair for $12 apiece. It cost $50 for a booth at the fair, and the ingredients for each cake cost $2. At some point, the club members will sell enough cakes so that their sales cover their expenditures. How much will the sales and expenditures be? Write a system of equations, graph them, and type the solution.

Explore how to determine the break-even point for a cooking club selling cakes at a street fair. Given a selling price of $12 per cake, a booth cost of $50, and an ingredient cost of $2 per cake, this problem demonstrates cost-revenue analysis and solving a system of linear equations. Suitable for Grades 7-9.

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