Atheater charges $25 for adults and $15 for children. when the price of a child's ticket increases to $18 next year, the cost for a dance club to attend the theater will increase from $450 to $480. write and solve a system of equations to find how many adults are in the dance club.

Given:
tickets for adults - 25
tickets for kids - 15
initial cost - 450
increase in tickets for kids to - 18
cost after increase - 480

let x be number of adults and y be number of kids

25x + 15y = 450
25x + 18y = 480

subtraction:

25x + 15y = 450
-25x - 18y = -480
 0      - 3y = - 30
         y = -30/-3
         y = 10

There are 10 kids.

25x + 15y = 450
25x + 15(10) = 450
25x + 150 = 450
25x = 450 - 150
25x = 300
x = 300/25
x = 12

There are 12 adults.

All in all there are 12 adults and 10 kids.

to check:
25x + 15y = 450
25(12) + 15(10) = 450
300 + 150 = 450
450 = 450

25x + 18y = 480
25(12) + 18(10) = 480
300 + 180 = 480
480 = 480