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The image contains a problem related to **systems of linear equations and inequalities**. Let's break it down.

The problem asks you to:

### Part A:
- Write the equation for the **cost function** and the **revenue function** based on a graph provided for the business "Muffins A-plenty" over the first six months.

From the graph:
- **Cost function** (appears to be linear): The cost at 0 months (the y-intercept) seems to be 55, and the cost at 6 months is 85.
- **Revenue function** (appears to be linear): The revenue starts at 0 (at 0 months) and reaches 60 at 6 months.

Using these values, we can calculate the slope of each line and form the equations of the functions.

### Cost Function \( y = mx + b \):
- The slope \( m \) can be found using two points: \( (0, 55) \) and \( (6, 85) \).
  \[
  m = \frac{85 - 55}{6 - 0} = \frac{30}{6} = 5
  \]
- The equation for the cost function is:
  \[
  y = 5x + 55
  \]

### Revenue Function \( y = mx + b \):
- Using the points \( (0, 0) \) and \( (6, 60) \):
  \[
  m = \frac{60 - 0}{6 - 0} = 10
  \]
- The equation for the revenue function is:
  \[
  y = 10x
  \]

### Part B:
- The **profit function** is the difference between the revenue and cost functions:
  \[
  \text{Profit function: } P(x) = \text{Revenue} - \text{Cost} = (10x) - (5x + 55)
  \]
  Simplifying:
  \[
  P(x) = 5x - 55
  \]
- To determine when the business makes its first profit, we set \( P(x) = 0 \):
  \[
  5x - 55 = 0
  \]
  Solving for \( x \):
  \[
  x = \frac{55}{5} = 11
  \]
  So, the business will make its first profit after **11 months** of operation.

### Part C:
- To predict the **profit after 10 months**, substitute \( x = 10 \) into the profit function:
  \[
  P(10) = 5(10) - 55 = 50 - 55 = -5
  \]
  The profit after 10 months is **-5 dollars**, meaning the business is still operating at a loss after 10 months.

If you have any further questions or need clarification on any part, feel free to ask!

### Relative Questions:
1. How would the equations change if the cost per month increased?
2. What happens if the revenue per month decreases over time?
3. How do you calculate the break-even point using a system of equations?
4. What are the implications of a negative slope in the revenue or cost function?
5. How would you model the impact of a one-time fee in addition to monthly costs?

### Tip:
When working with linear functions, always calculate the slope first to ensure accuracy in defining the relationship between variables.

The cost of a business is the amount of money spent to start the business and keep it in operation. The revenue is the amount of money the business earns from the sale of goods or services. Write the equation to represent the cost function and the revenue function for this business, and determine the profit after 10 months.

Learn how to solve a business-related problem by writing equations for cost and revenue functions. This example covers linear equations, slope-intercept form, and finding profit after 10 months. Suitable for students in Grades 9-12.

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