# Geometric interpretation of a Linear problem with bounded variables

I have a question of how to make a geometric interpretation of this problem

$$\begin{eqnarray} \mbox{max} & z = 3x_1+x_3 \\ s.a: & \\ & \begin{array}{cc} x_1+2x_2+x_3+x_4& =10 \\ x_1-2x_2+2x_3&=6\\ 1 \leq x_1\leq 4&\\ -5 \leq x_2 \leq 4\\ 0 \leq x_3 \leq 4&\\ 0 \leq x_4 \leq 5& \end{array} \end{eqnarray}$$

• How about solve for $x_3$ and $x_4$ explicitly (through the equality constraints), then plot the resulting inequalities as a polytope and the objective function as the level sets – Richard Jul 1 '19 at 19:23
• The question is clear, but I have some questions about the title. What does PL stand for? Is this a misspelling of LP? Also, I am not familiar with the term "channeled variables". – Kevin Dalmeijer Jul 1 '19 at 20:02
• "Programação linear com varíaveis canalizadas" in portuguese translates to "Linear programming with bounded with bounded variables" – Marcus Ritt Jul 2 '19 at 10:26
• Sry, typo, only "Linear programming with bounded variables". – Marcus Ritt Jul 2 '19 at 11:23
• Perhaps this is clear to others, but it is not clear to me what a "geometric interpretation" of an LP is. Can you clarify or provide a reference if this is an often used term? – Discrete lizard Jul 2 '19 at 11:30

From the first two constraints you will cancel out $$x_2$$ which gives you: $$2x_1 + 2x_3 + x_4 = 16$$. Now, you can solve for $$x_4$$ and substitute that in the last constraint ($$0 \le x_4 \le 5$$) and you will have a system which only relies on $$x_1$$ and $$x_3$$