To write an equation representing the total cost \( y \) as a function of the number of hours spent fixing a leak \( x \), we need to determine the fixed travel fee and the hourly cost based on the information provided.
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Identify the total cost for each scenario:
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For 2 hours, the total cost is $100: \[ y = \text{Travel fee} + (\text{Hourly fee}) \times 2 \] \[ 100 = \text{Travel fee} + 2h \]
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For 6 hours, the total cost is $260: \[ y = \text{Travel fee} + (\text{Hourly fee}) \times 6 \] \[ 260 = \text{Travel fee} + 6h \]
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Set up the equations using \( t \) for the travel fee and \( h \) for the hourly fee:
From the first equation: \[ 100 = t + 2h \quad \text{(1)} \]
From the second equation: \[ 260 = t + 6h \quad \text{(2)} \]
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Solve the system of equations:
Subtract equation (1) from equation (2): \[ 260 - 100 = (t + 6h) - (t + 2h) \] \[ 160 = 4h \] \[ h = 40 \]
Now substitute \( h = 40 \) back into equation (1): \[ 100 = t + 2(40) \] \[ 100 = t + 80 \] \[ t = 20 \]
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Formulate the total cost equation:
Now that we have the values for \( t \) and \( h \):
- Travel fee \( t = 20 \)
- Hourly fee \( h = 40 \)
The total cost equation \( y \) in terms of hours \( x \) is: \[ y = 20 + 40x \]
So, the final equation representing the total cost of a repair as a function of the number of hours spent fixing a leak is: \[ y = 20 + 40x \]