Question

Algebra unit 6 retake
A community arena is planning to host a concert. The arena has 250 seats available and sells two types of tickets. General admission tickets (G) and VIP tickets ( V) . The arena wants to find a combination of ticket sales that exactly fills the venue. Each General admission ticket sold allows for 1 attendee, while each VIP ticket sold allowance for 3 attendees.
Part 1. Create a 2 variable linear equation that represents the total number of attendees when all the seats filled. Clearly define your variables.
Part 2. The theater sells 80 General admission tickets and 50 VIP tickets. Determine if the ticket sales decision exactly fills the arena. Show your work and state weather the arena is filled with the given scenario.

Answer 1

Let's break down the problem step by step.

### Part 1: Creating the Linear Equation

Let:
- \( G \) = the number of General admission tickets sold
- \( V \) = the number of VIP tickets sold

From the problem, each General admission ticket allows for 1 attendee, while each VIP ticket allows for 3 attendees. Given that the arena has a total of 250 seats, we can create the following equation to represent the total number of attendees:

\[
G + 3V = 250
\]

### Part 2: Determining if the Ticket Sales Fill the Arena

In this scenario, the theater sells:
- \( G = 80 \) (General admission tickets)
- \( V = 50 \) (VIP tickets)

We will substitute these values into our equation to determine if the arena is filled.

Plug in the values:

\[
80 + 3(50) = 80 + 150 = 230
\]

Now we compare the total number of attendees to the total seats available:

**Total attendees:** 230
**Total seats available:** 250

Since \( 230 < 250 \), the ticket sales do not exactly fill the arena.

### Conclusion

The arena is **not filled** with the given scenario, as the total number of attendees (230) is less than the total seats available (250).