Question

A repair person charges a travel fee to visit a home and an hourly fee for time spent fixing a leak. A repair person that takes 2h charges $100. A repair that takes 6h cost $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.

Answer 1

To create an equation that represents the total cost of a repair, \( y \), as a function of the number of hours spent fixing a leak, \( x \), we need to identify the components of the cost.

We know from the problem:

1. A repair person charges a travel fee (let's call it \( F \)) and an hourly fee (let's call it \( H \)) for the work done.
2. From the first scenario, where the total cost for 2 hours is $100, we can write the equation:
\[
F + 2H = 100
\]
3. From the second scenario, where the total cost for 6 hours is $260, we can write the second equation:
\[
F + 6H = 260
\]

Now, we can solve this system of equations to find the values of \( F \) and \( H \).

**Step 1:** Subtract the first equation from the second equation to eliminate \( F \):
\[
(F + 6H) - (F + 2H) = 260 - 100
\]
This simplifies to:
\[
4H = 160
\]
So,
\[
H = 40
\]

**Step 2:** Substitute \( H \) back into one of the original equations to solve for \( F \). We can use the first equation:
\[
F + 2(40) = 100
\]
\[
F + 80 = 100
\]
\[
F = 20
\]

Now that we have \( F = 20 \) and \( H = 40 \), we can express the total cost \( y \) as a function of the number of hours \( x \):

The total cost \( y \) can be represented by the equation:
\[
y = F + Hx
\]
Substituting the values of \( F \) and \( H \):
\[
y = 20 + 40x
\]

Thus, the equation representing the total cost of a repair \( y \) as a function of the number of hours spent fixing a leak \( x \) is:
\[
y = 20 + 40x
\]