How to find nonnegative solutions of a linear system? I have the following system of $M$ linear equations in $N$ unknowns.
$$
\begin{bmatrix}
3 & 0 & 1 & 0 & -1 & -3  & 2\\
1 & 2 & 0 & 4 & 0 & 0  & -1\\
1 & 1 & 0 & 0 & -1 & -1  & -2\\
0 & 0 & 1 & 0 & -3 & -1  & 1 \\
\end{bmatrix}
\begin{bmatrix}
x_{1}\\
x_{2}\\
x_{3}\\
x_{4} \\
x_{5} \\
x_{6} \\
x_{7} \\
\end{bmatrix} =
\begin{bmatrix}
1\\
0\\
0\\
-1\\
\end{bmatrix}$$
Is there any algorithm for finding answers of this equations that ${x_{i} \ge 0}$?
Comment: I just want that $x_i \ge 0$.
It can change to
$$
\begin{bmatrix}
1 & 0 & 0 & 0 & 2/3 & -2/3 & 1/3 & 2/3\\
0 & 1 & 0 & 0 & -5/3 & -1/3 & -7/3 & -2/3 \\
0 & 0 & 1 & 0 & -3 & -1 & 1 & -1 \\
0 & 0 & 0 & 1 & 2/3 & 1/3 & 5/6 & 1/6 \\
\end{bmatrix}
$$

You can solve the linear programming problem given below by the simplex method:
$\max z=0$ subject to the constraints given by the equations.
Of course, you would have to add artificial variables for all the constraints which would make it a far too large problem to be solved by hand. Probably some software would be of assistance.
Wolfe's method in quadratic programming relies on the same idea for finding non negative solutions of the Kuhn Tucker conditions.
