Use a system of linear equations with two variables and two equations to solve. A concert manager counted 675 ticket receipts the day after a concert. The price for a student ticket was $\$ 12.50$, and the price for an adult ticket was $\$ 18.00$. The register confirms that $\$ 10,912.50$ was taken in. How many student tickets and adult tickets were sold?

Step 1 ：Let's denote the number of student tickets as x and the number of adult tickets as y. We then have two equations:

Step 2 ：1. \(x + y = 675\) (This is because the total number of tickets sold is 675)

Step 3 ：2. \(12.5x + 18y = 10912.5\) (This is because the total amount of money taken in is $10,912.50)

Step 4 ：We can solve this system of equations to find the values of x and y.

Step 5 ：The solution to the system of equations is x = 225 and y = 450. This means that 225 student tickets and 450 adult tickets were sold.

Step 6 ：Final Answer: The number of student tickets sold is \(\boxed{225}\) and the number of adult tickets sold is \(\boxed{450}\).