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 $\$ 40.00$. What time did they come home? They came home at A.M.

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 ：We know that \(x + y = 6\) (since they were out for 6 hours) and \(5x + 7.5y = 40\) (since they paid the babysitter $40).

Step 3 ：We can solve this system of equations using Gaussian elimination.

Step 4 ：By solving the system of equations, we get the values of x and y, which represent the number of hours before and after 11 P.M. respectively.

Step 5 ：We can then use these values to determine what time they came home.

Step 6 ：Final Answer: They came home at \(\boxed{3:00 \text{ A.M.}}\).