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

Let's define the variables:

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

From the problem statement, we know the following:

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

Since there are 300 seats available in the theater, we can set up the equation that represents the total number of attendees as follows:

\[ S + 2P = 300 \]

This equation indicates that the total number of attendees (the sum of those who have standard tickets and twice the number of those who have premium tickets) must equal the total number of seats available in the theater.

Part 2

In this scenario, the theater sells 100 standard tickets and 100 premium tickets. We need to find out if this combination fills the theater.

First, we can set \( S = 100 \) and \( P = 100 \). Now, we substitute these values into the equation we derived earlier:

\[ S + 2P = 300 \]

Substituting our values:

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

Now we can calculate:

\[ 100 + 200 = 300 \]

This simplifies to:

\[ 300 = 300 \]

Since both sides of the equation are equal, this means that when the theater sells 100 standard tickets and 100 premium tickets, it indeed fills the theater exactly.

Conclusion

The theater is filled with this ticket sales decision of 100 standard tickets and 100 premium tickets.