Use Gaussian elimination to solve. The Burkes pay their babysitter $\$ 5$ per hour before 11 P.M. and $\$ 7.50$ after 11 P.M. One evening they went out for $6 \mathrm{hr}$ and paid the sitter $\$ 42.50$. What time did they come home?

Step 1 ：Let's denote the number of hours before 11 P.M. as x and the number of hours after 11 P.M. as y.

Step 2 ：From the problem, we know that the total number of hours is 6, so we have the equation \(x + y = 6\).

Step 3 ：We also know that the total payment is $42.50, and the payment before 11 P.M. is $5 per hour and after 11 P.M. is $7.50 per hour. So we have the equation \(5x + 7.5y = 42.5\).

Step 4 ：Now we can use Gaussian elimination to solve this system of equations.

Step 5 ：The solution gives us the number of hours before and after 11 P.M.

Step 6 ：The Burkes came home at \(\boxed{2:30 \text{ A.M.}}\). This is because they spent x hours before 11 P.M. and y hours after 11 P.M. Since x and y are the number of hours, we can convert them to time. If x is 4.5, that means they spent 4.5 hours before 11 P.M., so they started at 6:30 P.M. (because 11 P.M. minus 4.5 hours is 6:30 P.M.). And if y is 1.5, that means they spent 1.5 hours after 11 P.M., so they ended at 2:30 A.M. (because 11 P.M. plus 1.5 hours is 2:30 A.M.).