(1)) A repair person charges a travel fee to visit a home and an hourly fee for the time spent fixing a leak. A repair that takes 2 h costs \( \$ 100 \). A repair that takes 6 h costs \( \$ 260 \). Write an equation to represent the total cost of a repair, \( y \), as a function of the number of hours spent fixing a leak, \( x \).

To represent the total cost of a repair as a function of hours spent fixing a leak, we can first define the costs involved. Let's denote the travel fee as \( T \) and the hourly fee as \( H \). From the first scenario, where a 2-hour repair costs \( \$100 \): \[ y = T + 2H = 100 \] From the second scenario, where a 6-hour repair costs \( \$260 \): \[ y = T + 6H = 260 \] Now we can solve these two equations simultaneously to express \( y \) in terms of \( x \). 1. From the first equation, we have \( T = 100 - 2H \). 2. Substituting \( T \) into the second equation: \[ 100 - 2H + 6H = 260 \] \[ 4H = 160 \quad \Rightarrow \quad H = 40 \] Now, substituting \( H \) back to find \( T \): \[ T = 100 - 2(40) = 20 \] So the travel fee \( T \) is \( \$20 \) and the hourly fee \( H \) is \( \$40 \). The equation representing total cost \( y \) as a function of hours \( x \) is: \[ y = 20 + 40x \] Now, what fun facts can I share about this? Travel fees in the repair business are often necessary to cover the repair person’s time to get to you, which means they might be spending as much time traveling as they do fixing—that’s why they exist! And if you've ever needed a plumber, you know that it helps to get that part sorted out upfront! Speaking of plumbing, did you know that the modern flush toilet was invented in the late 16th century? Sir John Harington was the inventor, and his design was not widely accepted until a few centuries later. So, next time you call a repair person, remember that the plumbing we often take for granted has a rich history!