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.