Why is $Ax \leq b$ is equivalent to $A'x' = b'$ and $x'\geq 0$

Question

I am getting confused in LP conversion from one form to another. Say, I have an LP of the form $Ax\leq b$ and I want to convert it into the form $A'x'=b', x'\geq 0$, how to do this?

For example let's say the equation is $3x+2y\leq 10$, we can introduce an dummy variable and convert it into $3x+2y+z = 10, z\geq 0$ but the form says that $x,y\geq 0$, how to enforce this constraint?

Answer 1

You also need to replace $x$ with $x^+ - x^-$, where $x^+ \ge 0$ and $x^- \ge 0$, with similar changes for $y$.