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)

GPT-4o mini · Answer

### Part 1: Create a two-variable linear equation **Variables:** - Let $ S $ be the number of standard tickets sold. - Let $ P $ be 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 states that the total number of attendees (from both types of tickets) must equal the total number of seats available (300). --- ### Part 2: Determine if the ticket sales decision exactly fills the theater **Given Scenario:** - Standard tickets sold: $ S = 100 $ - Premium tickets sold: $ P = 100 $ **Calculation:** Now, we can substitute these values into the equation we created: $$ 100 + 2(100) = 100 + 200 = 300 $$ The total number of attendees based on this ticket sale would be 300. **Conclusion:** Since the total number of attendees (300) equals the number of seats available (300), the theater is filled with the given scenario of selling 100 standard tickets and 100 premium tickets.