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The problem in the image involves a system of equations for finding the price of adult and student tickets based on two different scenarios.

### Problem Breakdown:
1. **Scenario 1**: Three adults and six students paid $102 in total.
   - Let the price of an adult ticket be **x** dollars.
   - Let the price of a student ticket be **y** dollars.
   - This gives us the equation:  
     \[
     3x + 6y = 102
     \]
   
2. **Scenario 2**: Two adults and seven students paid $95 in total.
   - This gives the equation:  
     \[
     2x + 7y = 95
     \]

### Step 1: Determine which system of equations matches
Among the provided options, option **C** matches the system of equations:
\[
\text{C: } 2x + 7y = 95 \quad \text{and} \quad 3x + 6y = 102
\]
Thus, the correct answer to the first part is **C**.

### Step 2: Solve the system to check Marcus' solution
Now, let's solve the system of equations:
1. **Equations**:
   \[
   3x + 6y = 102 \quad \text{(1)}
   \]
   \[
   2x + 7y = 95 \quad \text{(2)}
   \]

2. **Solve equation (1) for $x$** in terms of $y$:
   Divide the first equation by 3:
   \[
   x + 2y = 34 \quad \text{(3)}
   \]

3. **Substitute** equation (3) into equation (2):
   Replace $x$ with $34 - 2y$ in the second equation:
   \[
   2(34 - 2y) + 7y = 95
   \]
   Simplify:
   \[
   68 - 4y + 7y = 95
   \]
   \[
   68 + 3y = 95
   \]
   \[
   3y = 95 - 68 = 27
   \]
   \[
   y = 9
   \]

4. **Substitute $y = 9$ into equation (3)**:
   \[
   x + 2(9) = 34
   \]
   \[
   x + 18 = 34
   \]
   \[
   x = 16
   \]

Thus, the adult ticket costs **$16** and the student ticket costs **$9**.

### Step 3: Analyze Marcus' solution
Marcus found that the student ticket cost $12 and the adult ticket cost $9. This is incorrect because we found the correct prices to be:
- Adult ticket: $16
- Student ticket: $9

Therefore, the correct answer to part 4b is **B**:
- **B. No, Marcus did not solve the system of equations correctly. Each adult ticket was $16, and each student ticket was $9.**

### Summary:
- For the system of equations question, the correct option is **C**.
- For Marcus' solution, the correct option is **B**.

Would you like further details on how the system was solved or any other aspect of the problem?

Here are 5 related questions:
1. How do you solve a system of equations using substitution?
2. How would the problem change if more adults or students attended?
3. How can you solve a system of equations using elimination?
4. What happens if both equations represent parallel lines?
5. Can you solve this system graphically, and what would that look like?

**Tip**: Always double-check your work by substituting your solution back into the original equations to verify accuracy.

A youth group and their leaders visited a museum. Three adults and six students 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? Marcus wrote a system of linear equations to find the price for each adult and student. He found that each student ticket was $12 and each adult ticket was $9. Is Marcus correct? Explain.

Learn how to solve a system of equations to determine the price of adult and student tickets at a museum. This problem involves real-world algebra concepts, including systems of equations and substitution methods. Suitable for students in Grades 8-10.

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