I know that I must be wrong, but I'm hoping someone may be able to tell me where my understanding of the problem breaks down in the following question. If we consider the primal problem,
$$\min c^{T}x$$ $$\text{subject to } Ax=b$$
and its corresponding dual problem,
$$\max b^{T}y$$ $$\text{subject to } A^{T}y+s=c$$
then the following optimization problem can be solved by solving the following system of equations,
$$Ax-b=0$$ $$A^{T}y+s-c=0$$ $$b^{T}y-c^{T}x=0$$
where the final equation states that the duality gap must be zero. Wouldn't such a problem easily be solved by some simple linear algebra, since we have 3 unknowns $(x,y,s)$ and 3 equations?