How to minimize $|x_1| + |x_2|$ via linear programming? Given $a_{ij}, b_j \in \Bbb R$, consider the following optimization problem:
$$\begin{array}{ll} \underset{x_1, x_2 \in \Bbb R}{\text{minimize}} & |x_1| + |x_2|\\ \text{subject to} & a_{11}x_1 + a_{12}x_2 = b_1\\ & a_{21}x_1 + a_{22}x_2  = b_2\end{array}$$
Can I solve this problem with linear programming methods? If so, how?

You can!
Set $x_1=x_1^+ - x_1^-$ where $x_1^+ \geq 0$ and $x_1^- \geq 0$ then $|x_1|=x_1^+ + x_1^-$. Same for $x_2$.
This works since it is a minimization problem.
