how to solve a simplex with n variables I don't know how to resolve a simplex with $n$ variables 
I have this primal problem
\begin{cases}
\text{min}& z=-x_1 - x_2  -\cdots - x_n\\
&a_1x_1 + a_2x_2 +\cdots   + a_nx_n \le 1\\
&x_1,\dots, x_n \ge 0\\
&a_1,\dots, a_n \gt 0
\end{cases}
I found that the dual problem is 
\begin{cases}
\text{max}& w=y_1 + y_2  +\cdots + y_n\\
&a_1y_1  \ge -1\\
&a_2y_1  \ge -1\\
&\vdots\\
&a_ny_1  \ge -1\\
&y_1,\dots, y_n \ge 0\\
&a_1,\dots,a_n \gt 0
\end{cases}
Do I need to use simplex method or Complementary Slackness Theorem to solve this problem ? 
My first guess what to write :
$(a_1+a_2+...+a_n)y_1 \ge -n$ 
then I have $y_1 \ge \dfrac{-1}{(a_1+a_2+...+a_n)}$ 
but it doesn't solve the problem. 
If I add the slack variable I think we could see a solution but the obvious solution is that $y_1 = 0$ ...
If someone can explain me how to find the solution I would be grateful
 A: Method 1
Actually, the Fundamental Theorem of Linear Programming (which asserts that for every feasible (optimal) solution, there exists a basic feasible (optimal) solution) will give you the answer.  Using the Strong Duality Theorem and working on the dual problem will give you similar procedures and the same answer.  If the primal problem is correct, then a basic solution will be $x_i = \dfrac1{a_i}$, which gives the objective function value $-\dfrac1{a_i}$, so the optimal value for this minimization problem will be $\min\left\{-\dfrac1{a_i}:1\le i\le n\right\}$.
Method 2
If you insist on finding the dual problem, then it should be
$\max w = y_1$ subject to
\begin{align}
a_1 y_1 &\le -1 \\
a_2 y_1 &\le -1 \\
&\vdots \\
a_n y_1 &\le -1 \\
y_1&\le 0
\end{align}
That is to $\max w = -y_1$ subject to
\begin{align}
a_1 y_1 &\ge 1 \\
a_2 y_1 &\ge 1 \\
&\vdots \\
a_n y_1 &\ge 1 \\
y_1&\ge 0
\end{align}
Then $-y_i \le -\dfrac1{a_i} \;\forall 1 \le i \le n \iff -y_i \le \min \left\{-\dfrac1{a_i}:1\le i\le n\right\}$, so the result is still the same.
Comparison of the two methods
Of course using the first method is faster: you just substitute the numbers to get a basic solution.  Using the second method, one needs to write down the dual.  Then you need to handle those "forall" stuff before getting the answer.
