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.
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

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

**Creating the Linear Equation:**
Since each standard ticket allows for 1 attendee and each premium ticket allows for 2 attendees, the total number of attendees can be expressed by the equation:
\[
S + 2P = 300
\]
This equation represents the scenario where all 300 seats in the theater are filled.

### Part 2

**Given Sales:**
- The theater sells 100 standard tickets and 100 premium tickets.

**Calculating Total Attendees:**
Using the defined variables and plugging in the values:
- \( S = 100 \) (standard tickets)
- \( P = 100 \) (premium tickets)

Substituting these values into the equation:
\[
S + 2P = 100 + 2(100) = 100 + 200 = 300
\]

**Conclusion:**
The total number of attendees when selling 100 standard tickets and 100 premium tickets is 300.

**Is the theater filled?**
Yes, the theater is filled with this given scenario because the total number of attendees (300) matches the total seating available (300).