Linear relations in context
In a straight-line relation the rate of change is the slope, and the fixed amount is the value when x is zero.
A worked example
A 5 hour job costs $180 and a 12 hour job costs $362, so the rate is (362 - 180) / (12 - 5) = $26/h and the fixed fee is 180 - 5(26) = $50.
The mistake people actually make
Find the rate from the change in cost over the change in hours first. Dividing a whole bill by its hours mixes the fixed fee into the rate. The fixed fee is what remains once the hourly charge is taken off.