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Use Gaussian elimination to solve:

The Burkes pay their babysitter $5 per hour before 11 p.m. and $7.50 per hour after 11 p.m. One evening they went out for a total of 6 hours and paid the sitter $37.50. What time did they come home?

Answer :

Final answer:

By setting up two equations based on pay rates before and after 11 p.m. and the total time and amount paid, we use Gaussian elimination to find that the Burkes came home at 2 a.m.

Explanation:

To solve the question What time did the Burkes come home using Gaussian elimination, we need to set up a system of linear equations based on the given information about the babysitting rates and total time.

Let x represent the number of hours before 11 p.m. at $5 per hour and y represent the number of hours after 11 p.m. at $7.50 per hour.

The first equation is based on the total hours:

  • x + y = 6 (total hours they were out)

The second equation is based on the total amount paid:

  • 5x + 7.5y = 37.5 (total amount paid to the sitter)

We can now use Gaussian elimination to solve this system:

  1. Multiply the first equation by 5:
  • 5x + 5y = 30
Subtract this new equation from the second equation:
  • (5x + 7.5y = 37.5) - (5x + 5y = 30)
  • 2.5y = 7.5
  • y = 3
Substitute y = 3 back into the first equation:
  • x + 3 = 6
  • x = 3

Therefore, they spent 3 hours before 11 p.m. and 3 hours after 11 p.m. If they left at the beginning of the evening, they arrived home at 2 a.m.

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