matrix movie theater question

Question

a movie theater has a seating capacity of 1200 and charges $2 for children, $3 for students, and $4 for adults. at a screening with full attendance, there were one third as many adults as children and students combined. the receipts totaled $3500. how many adults attended the show?

Answer 1

Let x = number of children, y = number of students, and z = number of adults

x + y + z = 1200

2x + 3y + 4z = 3500

z = (1/3)(x + y)

(4/3)(x + y) = 1200

2x + 3y + 1200 = 3500

So, 2x + 3y = 2300

x + y = 900

Multiply the second equation by -2:

2x + 3y = 2300

-2x - 2y = -1800

Add the equations: y = 500

Back substitute to get x = 400, and z = 300