Why the dual and the canonical dual may have the same optimal solution?

Question

For instance let be

\begin{cases} \max & 3x_1 &+2x_2 &+4x_3 & -x_4\\ &x_1 &+x_2 &-2x_3&&\le4\\ &2x_1&+3x_3&&-4x_4&\ge 5\\ &2x_1&+x_2&+3x_3&&=7\\ x_1,x_2\ge 0\\ x_4\le 0 \end{cases}

Why may the dual and the canonical dual have the same optimal solution?

Answer 1

There is no duality gap in linear programming.

In the primal world, the original problem and the version where it is in canonical form share the same solution (as in objective function). Since there is no duality gap, the corresponding dual will achieve the same objective function as well.