A total of m approximate measurements are taken of the length $l$ of some object, and the results are: $l =l_1, l = l_2, ..., l = l_m$. Solve the resulting system of m equations with respect to one unknown $l$ in the sense of the least squares.
A Theoretical Introduction to Numerical Analysis, Ryaben'kii & Tsynkov
I'm using this textbook as a supplement to a class since I find it has more interesting (and tougher) questions. Here are my thoughts:
The system would look something like
$\begin{matrix} l + \epsilon_1 = l_1 \\ l + \epsilon_2 = l_2 \\ l + \epsilon_3 = l_3 \\ l + \epsilon_4 = l_4 \\ l + \epsilon_5 = l_5 \\ ... \\\ l + \epsilon_m = l_m \\ \end{matrix}$
How would I solve a system like this? I'm not sure what I'm looking for.
This problem is known as data validation or data reconcimiation.
You have $n$ measurements $(l_i,\sigma_i)$ and you look for the most probable value of $L$; it is given by the minimization of
$$F=\sum_{i=1}^n \left(\frac {l_i-L}{\sigma_i} \right)^2$$ so $$\frac{dF}{dL}=2\sum_{i=1}^n \frac {l_i-L}{\sigma_i^2}=0\implies L=\left(\sum_{i=1}^n \frac {l_i}{\sigma_i^2}\right)\left(\sum_{i=1}^n \frac {1}{\sigma_i^2}\right)^{-1}$$