Linear Regression Coefficients for Nonlinear Equation: Understanding the Relationship Between $a_0, a_1, c$, and $m$.

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I have this non linear equation $ y = mx/(c+x) $ and converting it to linear.
I converted this equation to linear as follows

$ 1/y = (c+x)/ mx $

$ 1/y = c/mx + x/mx $

$ 1/y = c/mx + 1/m $

I am using matrix approach to find coefficients $a_0$ and $a_1$ using my data set.
But my problem is how do I get the values of $c$ and $m$?

Is $a_0 = 1/m$
and

$a_1 = c/m$

or $a_1 = m$

I am getting confused that how to find the values of $c$ and $m$. Any help would be appreciated.

I also did the question in my notebook, image is attached for reference non linear to linear conversion to perform linear regression