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.

Question

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.

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.

Answer 1

Part 1

Let's define the variables:

• Let \( S \) represent the number of standard tickets sold.
• Let \( P \) represent the number of premium tickets sold.

Each standard ticket allows for 1 attendee. Each premium ticket allows for 2 attendees. Therefore, the total number of attendees \( T \) can be represented by the following equation:

\[ T = S + 2P \]

Since the theater has 300 seats available, we want the total number of attendees to equal 300 when all seats are filled. This gives us the equation:

\[ S + 2P = 300 \]

Part 2

Now, we need to check if selling 100 standard tickets and 100 premium tickets fills the theater:

Given:

• \( S = 100 \) (number of standard tickets sold)
• \( P = 100 \) (number of premium tickets sold)

We can substitute these values into our attendance equation:

\[ T = S + 2P \] \[ T = 100 + 2(100) \] \[ T = 100 + 200 \] \[ T = 300 \]

Therefore, the total number of attendees \( T \) is 300.

Since the total number of attendees is exactly equal to the number of available seats (300), we conclude that the theater is filled with this ticket sales decision.

Conclusion: Yes, the theater is filled with the sales of 100 standard tickets and 100 premium tickets since the total attendees equal the theater's capacity.