last tuesday, Regal Cinemas sold a total of 8000 movie tickets. proceeds totaled $57,000. tickets can be bought in one of 3 ways: a matinee admission costs $5, student admission is $6 all day, and regular admissions are $8. how many of each type of ticket was sold if twice as many student tickets were sold as matinee tickets?

The answer is that there were 1000 matinee tickets, 2000 student tickets, and 5000 regular tickets sold.

Step-by-step explanation:

Let m = number of Matinee tickets, s = number of student tickets, and r = number of regular tickets. The sum of these variables is equal to 8000:

m + s + r = 8000

Solve for r:

r = 8000 - m - s

There were twice the number of student tickets as matinee:

s = 2m

Substitute:

m + s + r = 8000

m + 2m + r = 8000

3m + r = 8000

r = 8000 - 3m

The number of each type of ticket sold at that price is equal to the total dollar amount:

5m + 6s + 8r = 57000

Substitute:

5m + 6 (2m) + 8 (8000 - 3m) = 57000

5m + 12m + 64000 - 24m = 57000

17m - 24m = - 7000

-7m = - 7000

m = 1000 Matinee tickets sold

s = 2m = 2 (1000) = 2000 Student tickets sold

r = 8000 - 3m = 8000 - 3 (1000) = 8000 - 3000 = 5000 Regular tickets sold.

Proof total number of tickets:

m + s + r = 8000

1000 + 2000 + 5000 = 8000

8000 = 8000

Proof dollar amount:

5m + 6s + 8r = 57000

5 (1000) + 6 (2000) + 8 (5000) = 57000

5000 + 12000 + 40000 = 57000

17000 + 40000 = 57000

57000 = 57000