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The problem provided involves solving a system of equations using the augmented matrix method. Here is a breakdown of the problem:

- Adult tickets cost $10.00 each.
- Senior tickets receive a discount of $2.00, meaning they cost $8.00 each.
- The theater sold 570 tickets in total.
- The total revenue from these tickets was $5070.

We need to determine how many adult tickets and how many senior tickets were sold.

### Define variables:
Let:
- $ x $ = the number of adult tickets sold
- $ y $ = the number of senior tickets sold

### Set up the system of equations:
1. The total number of tickets sold equation:
   \[
   x + y = 570
   \]
2. The total revenue equation:
   \[
   10x + 8y = 5070
   \]

### Augmented matrix form:
We will express this system of equations as an augmented matrix:

\[
\begin{pmatrix}
1 & 1 & | & 570 \\
10 & 8 & | & 5070
\end{pmatrix}
\]

Now, let's solve this using the augmented matrix method. I will calculate the solution.The solution to the system of equations is:

- $ x = 255 $ (number of adult tickets sold)
- $ y = 315 $ (number of senior tickets sold)

Thus, 255 adult tickets and 315 senior tickets were sold.

Would you like further details on how the matrix was reduced or any other clarifications?

Here are five related questions to further expand on the topic:
1. How does Gaussian elimination work step-by-step in solving systems of equations?
2. Can augmented matrices be used to solve more complex systems of equations involving three variables?
3. What happens if a system of equations has no solutions or infinitely many solutions when using matrix methods?
4. How can you use technology (e.g., calculators, software) to solve large augmented matrices?
5. What are the advantages of using matrix methods over substitution or elimination?

**Tip:** When solving systems of equations, always check the final solution by plugging the values back into the original equations.

A movie theater sells tickets for $10.00 each, with seniors receiving a discount of $2.00. One evening the theater sold 570 tickets and had revenue of $5070. How many of each type of ticket were sold? Solve the system of equations generated by this problem by using the augmented matrix method. Show the augmented matrix used as well as the answers.

Learn how to solve a system of linear equations using augmented matrices through a real-world example of calculating movie ticket sales. This problem involves setting up and solving equations to find the number of adult and senior tickets sold, applying concepts of linear algebra.

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