Fahrenheit and Celsius conversion – linear regression on perfect data

Fahrenheit and Celsius degrees are related in a linear way. Given a table with pairs of both Fahrenheit and Celsius degrees, we can estimate the constants to devise a conversion formula from degrees Fahrenheit to degrees Celsius, or vice versa:

⁰F	⁰C
5	-15
14	-10
23	-5
32	0
41	5
50	10

Analysis from first principles

We would like to derive a formula for converting F (degrees Fahrenheit) to C (degrees Celsius), as follows:

Here, a and b are the constants to be found. A graph of the  function is a straight line and, thus, is uniquely determined by two points. Therefore, we actually...