Answer: 194 children, 46 students and 97 adults attended.
Explanation:
Let x represent the number children that attended.
Let y represent the number students that attended.
Let z represent the number adults that attended.
The movie theater has a seating capacity of 337. Assuming it was filled to capacity, It means that
x + y + z = 337- - - - - - - - - - -1
The theater charges $5.00 for children, $7.00 for students, and $12.00 of adults. If the total ticket sales was $ 2456, it means that
5x + 7y + 12z = 2456- - - - - - - - -2
There are half as many adults as there are children. It means that
z = x/2
x = 2z
Substituting x = 2z into equation 1 and equation 2, it becomes
2z + y + z - 337
y + 3z = 337- - -- - - - - - -3
5 × 2z + 7y + 12z = 2456
10z + 7y + 12z = 2456
7y + 22z = 2456- - - - - - - -4
Multiplying equation 3 by 7 and equation 4 by 1, it becomes
7y + 21z = 2359
7y + 22z = 2456
Subtracting, it becomes
- z = - 97
z = 97
x = 2z = 2 × 97
x = 194
Substituting x = 194 and z = 97 into equation 1, it becomes
194 + y + 97 = 337
y + 291 = 337
y = 337 - 291
y = 46