Question

A local theater is planning to host a play. The theater has 300 seats available, and it sells two types of tickets: standard tickets (S) and premium tickets (P). The theater wants to find a combination of ticket sales that exactly fills the theater. Each standard ticket sold allows for 1 attendee, while each premium ticket sold allows for 2 attendees.
Part 1: Create a two-variable linear equation that represents the total number of attendees when all the seats are filled. Clearly define your variables. (5 Points)
Part 2: The theater sells 100 standard tickets and 100 premium tickets. Determine if this ticket sales decision exactly fills the theater. Show your work and state whether the theater is filled with this given scenario. (2 points)
(7 points)

Answer 1

### Part 1: Define Variables and Create the Equation

Let:
- \( S \) = the number of standard tickets sold
- \( P \) = the number of premium tickets sold

Since each standard ticket allows for 1 attendee and each premium ticket allows for 2 attendees, the total number of attendees can be expressed with the equation:

\[
S + 2P = 300
\]

This equation represents the total number of attendees filling all 300 seats when standard and premium tickets are sold.

### Part 2: Determine if Ticket Sales Fill the Theater

Given:
- Number of standard tickets \( S = 100 \)
- Number of premium tickets \( P = 100 \)

We can use the equation created in Part 1 to check if these ticket sales fill the theater:

\[
S + 2P = 300
\]

Substituting the values into the equation:

\[
100 + 2(100) = 300
\]

Calculating further:

\[
100 + 200 = 300
\]

Thus, we have:

\[
300 = 300
\]

### Conclusion
Since the equation holds true, the theater is indeed filled with the given scenario of selling 100 standard tickets and 100 premium tickets. Therefore, the ticket sales exactly fill the theater.