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. (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)

Answer 1

Part 1:

Let's define our variables:

• S: the number of standard tickets sold
• P: the number of premium tickets sold

The total number of attendees can be represented by the equation:

\[ S + 2P = 300 \]

This equation means that each standard ticket (S) allows for 1 attendee, and each premium ticket (P) allows for 2 attendees. The total number of attendees (from both types of tickets) must equal the total number of seats in the theater, which is 300.

Part 2:

The theater sells 100 standard tickets and 100 premium tickets. Let's determine the total number of attendees with this ticket sales decision.

1. Calculate the total number of attendees from standard tickets: \[ \text{Total from standard tickets} = S = 100 \]

2. Calculate the total number of attendees from premium tickets: \[ \text{Total from premium tickets} = 2P = 2 \times 100 = 200 \]

3. Calculate the total number of attendees: \[ \text{Total attendees} = S + 2P = 100 + 200 = 300 \]

Since the total number of attendees (300) equals the total number of seats available in the theater, the theater is filled with this given scenario.

Conclusion:

The theater is filled with the decision to sell 100 standard tickets and 100 premium tickets.