system of equation and matrice

Question

In order to choose the most appropriate musical to perform next, you visit the box office after the opening weekend. The theater you work for has a total of 500 seats and sells three different types of tickets:

1. adult tickets for $8
2. senior citizen tickets for$7
3. child tickets for $5

On Friday night, the box office states that they made $2,726 and that there was only 57 open seats left. After further discussion, one of the workers recalls that the box office sold twice as many child tickets as adult tickets.

Answer 1

The setup for this problem is as follows:

1. We choose variables: a for adults, s for seniors and c for child.
2. Then we have two relationships:

total number of tickets sold:

a + s + c = 500 - 57

total sold:

8a + 7s + 5c = 2756

The third relationship is the number of adult and child tickets.

2c = a.

Translating this into a system looks like this:

a + c + s = 443

8a + 5c + 7s = 2756

a - 2c = 0.

Trying to solve the above system, it turns out it is inconsistent, which means there is no solution that satisfies all the three requirements. However, if we flip the third requirement, twice as many adult tickets sold as child tickets sold ( 2a = c), you can get definite answer. (a = 115, c = 230, s = 98).

Corrected system is

a + c + s = 443

8a + 5c + 7s = 2756

2a - c = 0.

If you need help with the above system, let me know so I can provide step by step answer.