Linear Programming: from primal to dual system, how many variables needed? Suppose that you are trying to solve the optimization problem: 
$$
\begin{equation*}
\begin{array}{ll@{}ll}
\text{maximize}  & v \cdot x &\\
\text{subject to}& Ax \geq b    &A \in \mathbb{R}^{m\times n}
\end{array}
\end{equation*}
$$
(i.e. trying to solve an optimization problem in $n$ variables with $m$ linear inequality constraints).
This problem can be reduced to running a solution finding algorithm on a different system of linear equations in $k$ variables. What is the smallest value of $k$ for which this can be done. 

I have tried to say $n$, the number of variables we have but it turns out incorrect.
note: This question has been picked from Advanced Algorithms and Complexity Coursera course, week 2: Linear Programming.

The answer is $m+n$. The idea is that you need to solve for the given inequalities and for the dual as well. The given inequalities will have $n$ variables, the dual will have $m$, so all in all, you will need to solve for $m+n$ variables!
