Let's examine the given information step-by-step.
1. Understanding the Problem:
- Ervin sells vintage cars.
- He sells 13 cars every 3 months.
- We assume he sells cars at a constant rate.
- We need to determine the slope of the line that represents this linear relationship, with time in months on the [tex]\(x\)[/tex]-axis and the number of cars sold on the [tex]\(y\)[/tex]-axis.
2. What is a Slope in This Context?
- The slope (often denoted as [tex]\(m\)[/tex]) of a line in the context of these variables represents the rate of change in the number of cars sold per month.
3. Determining the Slope:
- The slope [tex]\(m\)[/tex] is calculated as the change in the [tex]\(y\)[/tex]-value (number of cars sold) divided by the change in the [tex]\(x\)[/tex]-value (time in months).
- Change in the number of cars sold = 13 cars (this is the vertical change, or rise).
- Change in time = 3 months (this is the horizontal change, or run).
4. Calculation of the Slope:
- Using the formula for slope:
[tex]\[
\text{slope} = \frac{\text{change in } y}{\text{change in } x} = \frac{13 \text{ cars}}{3 \text{ months}} = \frac{13}{3}
\][/tex]
5. Reviewing the Options:
- A. [tex]\(\frac{3}{13}\)[/tex] (This represents a smaller rate than needed, indicating less than one car per month, which is not correct given Ervin's rate.)
- B. 3 (This implies Ervin sells 3 cars per month, which is incorrect as he sells more than that per month.)
- C. [tex]\(\frac{13}{3}\)[/tex] (This represents selling approximately 4.33 cars per month and matches our calculated slope.)
- D. 13 (This implies selling 13 cars every month, while the reality is every three months.)
6. Correct Answer:
- The correct answer representing the rate of cars sold per month is:
[tex]\[
\frac{13}{3}
\][/tex]
Thus, the correct answer is [tex]\(C. \frac{13}{3}\)[/tex].
