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:
- 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.
- From the first scenario, where the total cost for 2 hours is $100, we can write the equation: \[ F + 2H = 100 \]
- 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: \[ y = 20 + 40x \]