a local theater is planning to host a play

Question

the theater has 300 seats available and it sells two types of tickets standard and premium th theatr wants to find a combo of ticket sales that exactly fills th theater each standard ticket sold allows for 1 attendee while each premium ticker sold allows for 2 attendees.

1. create a two varible linear equation that represnts the total number of attendees when all seats are filled.

2. the sell 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

To address the theater's ticket sales situation, we can set the problem up using a two-variable linear equation.

1. Create a Two-Variable Linear Equation

Let:

• \( s \) be the number of standard tickets sold.
• \( p \) be the number of premium tickets sold.

Each standard ticket allows for 1 attendee, and each premium ticket allows for 2 attendees. The total number of attendees should equal the total number of available seats, which is 300. This gives us the following equation:

\[ s + 2p = 300 \]

This equation represents the total number of attendees in the theater with all seats filled.

2. Ticket Sales Scenario Analysis

Now, let's analyze the ticket sales decision of selling 100 standard tickets and 100 premium tickets.

Using our variables:

• Standard tickets sold (\( s \)) = 100
• Premium tickets sold (\( p \)) = 100

Now, let's substitute \( s \) and \( p \) into the equation:

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

Conclusion

The total number of attendees based on this ticket sales scenario is 300, which is exactly the number of seats available in the theater.

Therefore, the theater is filled with this given scenario.