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 fils the theater. Show your work and state whether the theater is filled with this given scenario. (2 points)

Answer 1

Part 1: Define Variables and Create the Equation

Let's define our variables:

• \( S \): the number of standard tickets sold.
• \( P \): the number of premium tickets sold.

From the problem, we know that:

• Each standard ticket allows for 1 attendee.
• Each premium ticket allows for 2 attendees.

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

\[ S + 2P = 300 \]

This equation states that the total number of attendees (which is the sum of attendees from standard tickets and twice the number from premium tickets) must equal the total number of seats available (300).

Part 2: Check if the Given Ticket Sales Fill the Theater

According to the scenario presented, the theater sells:

• 100 standard tickets (\( S = 100 \))
• 100 premium tickets (\( P = 100 \))

We can substitute these values into our equation to check if the theater is filled:

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

Since the total number of attendees equals 300, we conclude that the theater is indeed filled with this ticket sales decision.

Thus, the answer is:

Yes, the theater is filled with the sale of 100 standard tickets and 100 premium tickets.