# Inverting Linear System of Inequalities

I have $$6$$ integral variables, $$m,z,p, m',z',p'$$. I have a set of three inequalities:

$$m\leq m' \leq p+m$$ $$m \leq m'-z' \leq p+m$$ $$2p+2m+z \leq 2m'+p'\leq 2p+2m+z$$ (The last one is an equality).

Or in Matrix format

$$\begin{bmatrix} m \\ m \\ 2m+2p+z \\ \end{bmatrix} \leq \begin{bmatrix} 1 & 0 & 0 \\ 1 & -1 & 0 \\ 2 & 0 & 1 \\ \end{bmatrix} \begin{bmatrix} m'\\ z'\\ p'\\ \end{bmatrix} \leq \begin{bmatrix} m+p\\ m+p\\ 2m+2p+z\\ \end{bmatrix}$$

I am trying to solve this system to bound $$m',z',p'$$ individually in terms of $$m,z,p$$. Meaning, I want an expression as: $${\bf f}(m,z,p)\leq \begin{bmatrix} m' \\ z' \\ p' \\ \end{bmatrix}\leq {\bf g}(m,z,p)$$ How can I invert this system?



Alternatively, I want a change of basis from $$m',z',p'$$ to an orthogonal basis $$l,m,n$$ such that $${\bf f}(m,z,p)\leq \begin{bmatrix} l \\ m \\ n \\ \end{bmatrix}\leq {\bf g}(m,z,p)$$ again holds.