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The problem asks to determine which system of equations can be used to find the price of an adult ticket $ x $ and a student ticket $ y $, given the following conditions:

- Three adults and six students paid $102.
- Two adults and seven students paid $95.

To model this situation, let's set up equations based on the two groups:

1. For the first group:
   - 3 adults and 6 students paid $102.
   - The equation will be: 
     \[
     3x + 6y = 102
     \]
   
2. For the second group:
   - 2 adults and 7 students paid $95.
   - The equation will be: 
     \[
     2x + 7y = 95
     \]

Now, let's check the options:

- Option A: $ 3x + 2y = 102 $ and $ 6x + 7y = 95 $ is incorrect because the first equation should have $ 6y $, not $ 2y $, and the second equation should have $ 2x $, not $ 6x $.
- Option B: $ 6x + 3y = 102 $ and $ 7x + 2y = 95 $ is incorrect because the coefficients of $ x $ and $ y $ are swapped.
- **Option C**: $ 3x + 6y = 102 $ and $ 2x + 7y = 95 $ is correct. Both equations match the setup of the problem.
- Option D: $ 3x + 6y = 102 $ and $ 2x + 7y = 95 $ is also correct, but matches Option C exactly.

Thus, the correct system of equations is provided in **Option C** (or Option D as it matches).

Would you like more details on how to solve this system or ask further questions?

Here are some related questions to deepen understanding:
1. How can you solve the system of equations using substitution?
2. How can you solve the system of equations using elimination?
3. How do you verify your solution once you've solved the system?
4. What happens to the system of equations if the number of students or adults changes?
5. How does the price of the tickets affect the system if it is increased by a certain percentage?

**Tip**: When forming equations from word problems, always pay attention to how variables relate to the given quantities.

A youth group and their leaders visited a museum. Three adults and six students in one van paid $102 for their admission. Two adults and seven students paid $95 for their admission. Which system of equations can be used to find the price in dollars of an adult ticket, x, and a student ticket, y?

Explore how to set up and solve a system of equations involving ticket prices for adults and students. Given two scenarios from a youth group's museum trip, learn how to find the cost of each ticket type. Includes detailed steps and solution verification methods.

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