How to use the Simplex-method for constraints which are given as equations? I got stuck with the solution of an optimization problem which I have to solve. I read a lot about how to use the simplex method  and I can successfully apply it when the constraints are of the type "<= " or "<" , but I don't manage to understand how to use the method when I have equations instead.

Could someone explain to me how to make my Simplex table and how to solve it step by step? Thank you very much !
 A: You can transform equality constraints to inequality constraints. A constraint like $x=5$ can be written as two constraints: $$x \leq 5 \\ x\geq5 $$.
These two constraints will force x to equal 5.
Your first constraint can thus be written as the following two inequality constraints:
$$ x_1 -2x_4 +4x_5 \leq 1 \\
 x_1 -2x_4 +4 x_5 \geq 1$$.
Multiplying the second constraint by -1 will bring it into the form $\leq$. 
A: Another way of looking at the problem is by noticing that the constraints are equivalent to
\begin{cases}
-2x_4+4x_5 &\le 1 \\
x_4 &\le 1 \\
3x_4+2x_5 &\le 4
\end{cases}
In other words, you can consider that $x_1,x_2,x_3$ are your slack variables (with non null coefficients in the objective function).
In this case, the initial solution consists in setting variables $x_4$ and $x_5$ to $0$, which yields $X_0 =(1,1,4,0,0)$ which is the initial solution that you proposed. 
Now, start your table simplex like you usually would. In the first iteration, $x_4$ is the entering variable (with reduced cost $10$), and $x_2$ exits. Can you take it from there?
