Questions: 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 4 hr and paid the sitter 27.50. What time did they come home?

Solution Steps

To solve this problem, we need to set up a system of linear equations based on the given information. Let \( x \) be the number of hours before 11 P.M. and \( y \) be the number of hours after 11 P.M. We know that the total time is 4 hours, and the total payment is $27.50. We can express these conditions as two equations: \( x + y = 4 \) and \( 5x + 7.5y = 27.5 \). We will use Gaussian elimination to solve this system of equations.

We are given that the Burkes pay their babysitter $5 per hour before 11 P.M. and $7.50 per hour after 11 P.M. They went out for a total of 4 hours and paid the sitter $27.50. Let \( x \) be the number of hours before 11 P.M. and \( y \) be the number of hours after 11 P.M. We can set up the following system of equations:

1. \( x + y = 4 \)
2. \( 5x + 7.5y = 27.5 \)

Using the method of Gaussian elimination, we solve the system of equations:

1. From equation 1: \( x = 4 - y \)

2. Substitute \( x = 4 - y \) into equation 2:

   \[ 5(4 - y) + 7.5y = 27.5 \]

   Simplifying, we get:

   \[ 20 - 5y + 7.5y = 27.5 \]

   \[ 2.5y = 7.5 \]

   \[ y = 3 \]

3. Substitute \( y = 3 \) back into equation 1:

   \[ x + 3 = 4 \]

   \[ x = 1 \]

Since \( y = 3 \) represents the hours after 11 P.M., they came home 3 hours after 11 P.M., which is at 2 A.M.

Final Answer