Update: The question has beenreopened.

I intended to answer 김종현's problem on Math.SE. However, the programs in
the question body *aren't* typeset in MathJax. As a result, I downvoted
and closed this question because found it *unclear*. From the proposed dual,
it seems that I *shouldn't* interpret the primal as a linear program. Anyways,
*without* further clarifications from OP, I *found* no reason to look at this
further. Here's my intended answer:

First, you have to properly write the primal as

\[ \begin{alignedat}{8} \max \quad & z = & 3w_1 & + & 4 w_2 & + & 5w_3 & && \\ \text{s.t.} \quad & & w_1 & - & w_2 & & & - & \varepsilon_1 & & & & & \le 0 && \\ & & & & w_2 & - & w_3 & & & - & \varepsilon_2 & & & \le 0 && \\ & & & & & & w_3 & & & & & - & \varepsilon_3 & \le 0&& \\ & & & & & & & & 2\varepsilon_1 & + & 3\varepsilon_2 & + & 4\varepsilon_3 &\ge 1 && \\ & & w_1 & + & w_2 & + & w_3 & & & & & & & = 1, && \end{alignedat} \]

Your claimed dual

$$ \begin{array}{ll} \max & \varepsilon_1 r_1 + \varepsilon_2 r_2 + \varepsilon_3 r_3 + (2\varepsilon_1 +3\varepsilon_2 + 4\varepsilon_3) r_4 + r_5 \\ \text{s.t. } & x_1 + x_5 \le 3 \\ & x_2 \le 4 \\ & x_3 \le 5 \\ & r_4, r_5 \text{ free} \end{array} $$is incorrect since it's

nota linear program.