A group of 5 adults and 3 children paid a total of $108 for their concert tickets. Another group of 3 adults and 10 children paid $155. Find the cost of an adult ticket and the cost of a child’s ticket.

To solve this problem, let's set up a system of linear equations based on the information given.

Let:
- $ x $ represent the cost of an adult ticket.
- $ y $ represent the cost of a child's ticket.

We can derive two equations from the information provided:

1. **First Group:** 5 adults and 3 children paid $108.
   \[
   5x + 3y = 108
   \]

2. **Second Group:** 3 adults and 10 children paid $155.
   \[
   3x + 10y = 155
   \]

Now, we have the following system of equations:
\[
\begin{aligned}
5x + 3y &= 108 \quad \text{(Equation 1)}\\
3x + 10y &= 155 \quad \text{(Equation 2)}
\end{aligned}
\]

### Solving the System of Equations

We can use either the substitution method or the elimination method. Let's use the elimination method:

1. **Multiply Equation 1 by 3 and Equation 2 by 5** (to make the coefficients of $ x $ equal):

   \[
   \begin{aligned}
   15x + 9y &= 324 \quad \text{(Equation 1 multiplied by 3)}\\
   15x + 50y &= 775 \quad \text{(Equation 2 multiplied by 5)}
   \end{aligned}
   \]

2. **Subtract the first new equation from the second:**

   \[
   (15x + 50y) - (15x + 9y) = 775 - 324
   \]

   This simplifies to:

   \[
   41y = 451
   \]

3. **Solve for $ y $:**

   \[
   y = \frac{451}{41} = 11
   \]

   So, the cost of a child’s ticket is **$11**.

4. **Substitute $ y = 11 $ back into one of the original equations (use Equation 1):**

   \[
   5x + 3(11) = 108
   \]

   Simplifying:

   \[
   5x + 33 = 108
   \]

   \[
   5x = 108 - 33
   \]

   \[
   5x = 75
   \]

   \[
   x = \frac{75}{5} = 15
   \]

   So, the cost of an adult ticket is **$15**.

### Final Answer:
- The cost of an adult ticket is **$15**.
- The cost of a child’s ticket is **$11**.

Would you like a detailed breakdown of any part of the solution? Here are some related questions you might explore:

1. How can you solve the system using the substitution method?
2. What happens if you change the group sizes? How does it affect the equations?
3. Can you create a similar problem with different prices?
4. What if a third group is introduced? How would that impact the solution?
5. Can the elimination method be applied to non-linear systems of equations?

**Tip:** Understanding different methods for solving systems of equations, like substitution and elimination, helps in choosing the most efficient approach for various problems.

Learn how to solve a system of linear equations to find the costs of adult and child concert tickets. This problem involves using mathematical concepts such as linear equations and systems of equations. Suitable for students in Grades 7-9.

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