https://file.aiquickdraw.com/mathbot/file/17265247759870o9dl72c.jpg

Let's define variables for the unknowns in the problem:

- Let $ a $ represent the number of adults.
- Let $ s $ represent the number of students.
- Let $ c $ represent the number of children.

From the problem, we have the following information:

1. **Total number of people**:
   \[
   a + s + c = 400
   \]
2. **Ticket prices**: 
   - Adults: \$40
   - Students: \$20
   - Children: \$10

   The total amount of money collected is \$10,600, so:
   \[
   40a + 20s + 10c = 10,600
   \]
3. **Relation between children's tickets and other tickets**: 
   The number of children's tickets is 200 less than the number of adult and student tickets combined:
   \[
   c = (a + s) - 200
   \]

We now have a system of three equations:

\[
\begin{aligned}
1. & \quad a + s + c = 400 \\
2. & \quad 40a + 20s + 10c = 10,600 \\
3. & \quad c = (a + s) - 200
\end{aligned}
\]

We can substitute equation (3) into equations (1) and (2) to eliminate $ c $, then solve for $ a $ and $ s $.

Let's proceed with solving the system.The solution gives us:

- $ a = 180 $ adults
- $ s = 120 $ students

Now, using the equation $ c = (a + s) - 200 $, we can calculate the number of children:

\[
c = (180 + 120) - 200 = 100
\]

Thus, the number of attendees is:
- 180 adults
- 120 students
- 100 children

Would you like more details or have any questions?

Here are 5 related questions:
1. What is the total revenue generated from just adult tickets?
2. If the ticket prices for children increased, how would that impact the total revenue?
3. How many more children's tickets would need to be sold to match the revenue from adult tickets?
4. What would happen to the total revenue if 50 more students attended instead of children?
5. How would the system change if the children's ticket prices were \$15 instead of \$10?

**Tip**: Always check that your solution satisfies all given conditions when solving systems of equations.

A concert audience of 400 people consists of adults, students, and children. The ticket prices are $40 for adults, $20 for students, and $10 for children. The total amount of money taken in is $10,600. The number of children's tickets sold is 200 less than the number of adult and student tickets in total. How many adults, students, and children are in attendance?

Learn how to solve a system of equations to determine the number of adults, students, and children in a concert audience. The problem involves a total audience of 400 people, varying ticket prices, and a total revenue of $10,600. Ideal for students in Grades 9-12, this solution uses algebraic methods such as substitution.

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