I am using a book called Introduction to Operations Research.
I'm not sure how to deal with free variables that are not constrained i.e. they could be positive or negative. I understand how any vector could be written as the difference of $2$ non negative vectors with one of the vectors having all components the same. However, I don't understand how to solve the linear program below:
\begin{align} \min &\quad2x_1 + 3x_2 - x_3 + x_4\\ \mathrm{s.t.}&\quad x_1 + 2x_2 - x_3 - 5x_4 = 10\\ &\quad -2x_1 + 3x_3-2x_4 = 5\\ &\quad x_1\geqslant0 \end{align}
You replace the free variables $x_j$ by $x_j^+-x_j^- \ \ \forall \ \ j \in \{2,3,4\}$.
Therefore the objective function is
$2x_1 + 3(x_2^+-x_2^-) - x_3^++x_3^- + x_4^+-x_4^-$
Similar transformations for the constraints. The domain for the variables is $x_1,x_2^+,x_2^-,x_3^+,x_3^-,x_4^+,x_4^- \geq 0$.
Now you can apply an algorithm, for instance the simplex algorithm.