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)

Answer 1

Part 1: Create a Two-Variable Linear Equation

Let:

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

Each standard ticket allows for 1 attendee, while each premium ticket allows for 2 attendees. We want to fill all 300 seats in the theater. Therefore, the equation that represents the total number of attendees is:

\[ S + 2P = 300 \]

Part 2: Determine if the Ticket Sales Decision Fills the Theater

Given:

• The theater sells 100 standard tickets (\( S = 100 \)) and 100 premium tickets (\( P = 100 \)).

Now, we can substitute these values into our equation to see if it fills the theater:

\[ 100 + 2 \times 100 = 100 + 200 = 300 \]

Since the total number of attendees is 300, which equals the total number of seats available in the theater, we conclude:

The theater is filled with this given scenario.