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: Create a two-variable linear equation

**Variables Definition**:
- Let \( S \) represent the number of standard tickets sold.
- Let \( P \) represent the number of premium tickets sold.

**Equation**:
Since each standard ticket allows for 1 attendee and each premium ticket allows for 2 attendees, the total number of attendees can be represented by the equation:

\[
S + 2P = 300
\]

This equation captures the condition that the total attendance must exactly fill the theater's 300 seats.

### Part 2: Determine if the ticket sales decision exactly fills the theater

The theater sells 100 standard tickets and 100 premium tickets. We need to substitute \( S \) and \( P \) into the equation from Part 1.

Given:
- \( S = 100 \)
- \( P = 100 \)

Substituting these values into the equation:

\[
100 + 2(100) = 100 + 200 = 300
\]

Since the left side of the equation equals 300, which matches the total number of available seats, we conclude:

**Conclusion**: The theater is filled with this given scenario.