Given the following equation:
$y = \frac{80-x}{100-x}$
If I had real world data that followed this exact equation, how would I back out an equation to always predict the correct $y$ value given an $x$ value?
A second degree polynomial comes close but is not exact. I would hope for the resulting equation to be simple, given the simplicity of the equation to generate the data.
Sample Data:
x, y
0.000, 0.800
10.000, 0.778
20.000, 0.750
30.000, 0.714
40.000, 0.667
50.000, 0.600
60.000, 0.500
70.000, 0.333
80.000, 0.000
$$y=\frac{ax+b}{x+c}\quad\implies\quad xy=-cy+ax+b$$ Approximate $a,b,c$ are computed thanks to linear regression :
The above method is valid because no criteria of fitting is specified in the question.
If a particular criteria of fitting was requiered non-linear regression should be necessary. But the result would be very close to the above result because the data is almost not scattered.