Can someone explain how to do the change of variables in a linear programming problem?

Question

If I want to convert the following linear problem into a standard form: $$\min z=4x_1+2x_2+x_3$$ Subject to $$-x_1+3x_2-x_3\ge1$$ $$5x_1+3x_3=5$$ $$x_1+x_2+x_3\le1$$ $$-1\le x_1, x_2\le2,x_3\ge0$$ I understand how to turn $\min z$ into $\max z$ and how to use the slack variables. What I don't get is how to do the change of variables for the constraints. In the answer I am suppose to introduce the variable $x_1'=x_1-1$, $x_2'=-x_2-2$. Why do we write it like this? What is the logic behind it? And what if the constraint is with the absolute value like: $|x_1|\le1$?

Answer 1

As suggested by the first comment, in the standard form, you have nonnegative decision variables. $\require{extpfeil} \Newextarrow{\xiff}{5,5}{0x21d4}$

\begin{alignat}{3} -1 &\le x_1 &\iff x_1 + 1 &\ge 0 &\quad \xiff[]{\large \text{set } x_1'=x_1+1} \quad x_1' &\ge 0 \\ x_2 &\le 2 &\iff -x_2 + 2 &\ge 0 &\quad \xiff[]{\large \text{set } x_2'=-x_2+2} \quad x_2' &\ge 0 \end{alignat}

If you have $|x_1| \le 1$, you have to split it into two inequalities and include them in the constraints in the canonical form.

$$|x_1| \le 1 \iff \begin{cases} x_1 &\le 1 \\ -x_1 &\le 1 \end{cases} $$