Change of Variables(nonlinear function to a linear one)

[HclGuy]

Homework Statement

I'm given a nonlinear model which I am trying to perform a linear regression on. I need to use change of variables in order to convert the nonlinear model into a linear one.

Homework Equations

I am given $$f(x) = \frac{ax}{b+x}$$ where a and b are just parameters.
I need to use change of varibles to change this function into a form of $$F(x) = AX + B$$

The Attempt at a Solution

I have tried to take the reciprocal of both sides to get
$$\frac{1}{y} = \frac{b+x}{ax} = \frac{b}{ax} + \frac{1}{a}$$ , only problem with that is that the x is in the denominator. I have also tried using log transformation to both sides without success. Any help is appreciated thanks!

 

[Random Variable]

see below

 

[Random Variable]

I would rewrite it as b*f(x) - ax = x*f(x)

or in matrix form

$$\left(\begin{array}{cc}x_{1}&f(x_{1})\\x_{2}&f(x{2})\\.&.\\.&.\\.&.\\x_{n}&f(x_{n})\end{array}\right) \left(\begin{array}{cc}-a\\b\end{array}\right) = \left(\begin{array}{cc}x{1}*f(x_{1})\\x{2}*f(x{2})\\.\\.\\.\\x_{n}*f(x_{n})\end{array}\right)$$