Apply the system of equations method to solve a ticket mixture application.

Problem: A science center sold $1{,}363$ tickets on a busy weekend. The receipts totaled $\$12{,}146$. How many $\$12$ adult tickets and how many $\$7$ child tickets were sold?

1. Name the unknowns. Let $a =$ the number of adult tickets and $c =$ the number of child tickets.
2. Translate into a system of equations.

• The total number of tickets sold is $1{,}363$: $a + c = 1{,}363$
• The total value of adult tickets is $12a$ and the total value of child tickets is $7c$. The total receipts are $\$12{,}146$: $12a + 7c = 12{,}146$ The system of equations is: $a + c = 1{,}363$ $12a + 7c = 12{,}146$

3. Solve the system using the elimination method. Multiply the first equation by $-7$: $-7(a + c) = -7(1{,}363)$ $-7a - 7c = -9{,}541$ Add this to the second equation: $12a + 7c = 12{,}146$ $5a = 2{,}605$ $a = 521$ Substitute $a = 521$ into the first equation: $521 + c = 1{,}363$ $c = 842$
4. Check the result. $521$ adult tickets at $\$12$ each is $\$6{,}252$. $842$ child tickets at $\$7$ each is $\$5{,}894$. $6{,}252 + 5{,}894 = 12{,}146$ $\checkmark$
5. Answer the question. The science center sold $521$ adult tickets and $842$ child tickets.