For the event to kings island the cost for an adult ticket is $20 and the cost for a student ticket is $10. the tickets are purchased in advance and a total of 550 tickets were purchased and the total cost of the tickets was $7,500.

identify the total number of adult tickets and student tickets sold that would create a total of 550 tickets sold for a total cost of $7,500.

We have two formulas:
Adult tickets plus Student tickets = 550 tickets   or   A + S = 550
   and
Adult tickets times $20 plus Student ticket times $10 = $7500  or
(A * 20) + (S * 10)  = 7500

First let's get rid of one of the variables by re-balancing the simplest formula
A+S = 550
A+S-A = 550-A
S = 550-A

Now we can restate the second formula replacing "S"  with it's equivalent "550-A"

(A * 20) + (S * 10)  = 7500
(A * 20) + ((550-A) *10) = 7500
Then do the multiplication
20A + 5500 - 10A = 7500
and simplifiy
20A + 5500 - 5500 - 10A = 7500 - 5500
20A - 10A = 2000
10A = 2000
A =  200   
S =  550 - 200 = 350

Now we should test the answers to be sure our work
 is correct:
  200 Adult tickets times $20 = $4000
  350 Student tickets times $10 = $3500
  $4000 + $3500 = 7500