If I understand correctly, you have a linear programming problem $P$ and a basic solution $x^*$ with corresponding basic solution $y^*$ of the dual problem $D$ such that $y^*$ is feasible for $D$ but $x^*$ is not feasible for $P$ due to a single violated inequality constraint $C_i$.  

Of course it's possible that there are no optimal solutions at all.  It is also possible that there are optimal solutions in which your constraint $C_i$ is not an equality.

Consider the rather trivial linear programming problem $P$:

maximize $0$

subject to 
$$ \eqalign{x_1 &\le 1\cr
            x_1 &\le 2\cr
         x_1 &\ge 0\cr}
$$
Let $s_1$ and $s_2$ be the slack variables corresponding to the two constraints. 
The basic solution for basis $x_1, s_1$ is $x_1 = 2$, $s_1 = -1$, $s_2 = 0$ which violates the first constraint.  All basic solutions of the dual are $y_1 = 0$, $y_2 = 0$ which is feasible.  There are optimal solutions $0 \le x_1 \le 1$.