Let's break down the problem step by step and derive the correct equation that represents the given scenario.
1. Understanding the Costs:
- The one-time registration fee is \[tex]$45. This fee is a fixed amount and does not change with the number of lessons.
- Each music lesson costs \$[/tex]9. This fee is variable and depends on the number of lessons Justine takes. If she takes [tex]\( x \)[/tex] lessons, the total cost for the lessons is [tex]\( 9x \)[/tex].
2. Total Cost Calculation:
- The total cost ([tex]\(C\)[/tex]) that Justine pays can be represented as the sum of the fixed registration fee and the cost of the [tex]\( x \)[/tex] lessons. Therefore,
[tex]\[
C = 45 + 9x
\][/tex]
3. Average Cost per Lesson:
- The average cost per lesson is calculated by dividing the total cost by the number of lessons ([tex]\( x \)[/tex]).
[tex]\[
\text{Average Cost} = \frac{C}{x} = \frac{45 + 9x}{x}
\][/tex]
4. Given Average Cost:
- It is given that the average cost per lesson is \$12. Hence, we set up the equation:
[tex]\[
12 = \frac{45 + 9x}{x}
\][/tex]
5. Examining the Options:
Let's compare the derived equation ([tex]\( 12 = \frac{45 + 9x}{x} \)[/tex]) with the given options:
- Option A: [tex]\( 12 = \frac{54}{x} \)[/tex] – This does not match our derived equation.
- Option C: [tex]\( 12 = \frac{45 + 9 \pi}{x} \)[/tex] – This does not match our derived equation.
- Option D: [tex]\( 12 = \frac{4: x + 9}{x} \)[/tex] – This does not match our derived equation.
It appears there isn't an option in the list that directly matches the correct derived equation [tex]\( 12 = \frac{45 + 9x}{x} \)[/tex].
However, based on our detailed calculations and understanding of the provided choices, it seems none of the options provided are correct representations of the scenario.
Please let me know if there are additional or corrected answer choices, or if you need further assistance!
