let the number of main floor tickets be m, the number of balcony tickets be b, the number of gallery tickets be g

an equation for the total cost of tickets is

35m + 25b + 15g = 13275 and since there were 475 tickets, you have

m+b+g = 475 and since there are 45 more main floor tickets than balcony tickets, you also have

m = b+45

substiute this into the second equation for m

(b+45)+b+g=475

2b + 45 + g = 475 solve this for g in terms of b

2b + g = 430

g = 430/2b

substitute this and the equation for m into the first equation and solve for b

35(b+45) + 25b + 15(430/2b) = 13275

35b + 1575 + 25b + 6450/2b = 13275

60b + 6540(1/2b) = 11700

60b + 3270b = 11700

3330b = 11700

b = 1

plug this into the equations for m and g

m = b+45 = 1 + 45 = 46

g = 430/2b = 430/2 = 215

Therefore, 46 main floor tickets, 1 balcony ticket, and 215 gallery tickets were sold.