I have been given a least squares problem to solve, with the solution but I do not understand the steps provided and I am seeking clarity on why certain things in the solution have been done. 

I am given a nonlinear model along with measurements for $T\_j, K\_j$ and $\Delta K\_j$.

$K\_j = C\exp(-E/T\_j)+R\_j$,

where $E$ and $C$ need to be estimated and $R\_j$ is independent and normally distributed. 

First I am asked to show the model can be transformed to an equation of a different form that is linear:

$\ln(K\_j)=\ln(C)-\frac{E}{T\_j}+r\_j$,

where $r\_j=\frac{R\_j}{C\exp(-E/T\_j)}$, which I can do. But after this I can't follow the solution very well.

I need to show that the standard deviation of $r\_j=\delta\_j=\frac{\Delta K\_j}{K\_j}$. The solution states because $r\_j$ is independent and normally distributed we can write:

$r\_j=\frac{R\_j}{K\_j-R\_j}\approx\frac{R\_j}{K\_j} \rightarrow \delta\_j:=\frac{K\_j}{K\_j}$.

So I don't really understand why this has been done, obviously I have a gap in my knowledge here and I am looking to fill it!

The problem then continues and states that in order to use the least squares method $r\_j$ needs to be scaled in such a way that all $r\_j$ have the same standard deviation and the following relation:

$\sum^m\_{j=1}\left(\frac{r\_j}{\delta\_j}\right)^2=\sum^m\_{j=1}\left(\frac{-E/T\_j+\ln(C)-\ln(K\_j)}{\delta\_j}\right)^2=\min$.

The solution continues in code, but that is ok. What I need to understand is how the problem is formulated and rearranged in this way. As I understood it we apply the least squares method when we can't solve a system but want to find the closest solution possible to solving a system. 

So specific questions I have regarding this choice of rearrangement would be:

Why do we make the choice to set $r\_j$ on one side and not one of the other variables such as $T\_j$ or $K\_j$?

Why is it important that the standard deviation is scaled to 1 for everything (I assume that is why we have divided by it)?

When I look at the problem I get a system where the form should be $||Ax-b||\_2=min$. Why do I make the choice that A will consist of the $\frac{1}{T\_j}$ and b to consist of $\ln(K\_j)$?

The main problem I think is I am used to solving this with artificial problems and this problem looks more "real" and has actual data but I am having trouble following the method which means I haven't understood it so well. I hope I have been clear and provided enough information, but please let me know if anything is confusing or if I could provide anything else to help you help me. Cheers.