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 choose three unknown values as variables, for example $x_5$, $x_6$ and $x_7$, then the remaining unknown values become a linear function of these variables. In this case,
$$
\begin{bmatrix}
x_1 \\
x_2 \\
x_3 \\
x_4
\end{bmatrix}
=\frac{1}{6}\left(
\begin{bmatrix}
4 \\
-4 \\
-6 \\
1
\end{bmatrix}
+
\begin{bmatrix}
-4 & 4 & -2 \\
10 & 2 & 14 \\
18 & 6 & -6 \\
-4 & -2 & -5
\end{bmatrix}
\begin{bmatrix}
x_5 \\
x_6 \\
x_7
\end{bmatrix}
\right)
$$
If you for example want to find which values for $x_5$, $x_6$ and $x_7$ can be used such that $x_3$ and $x_4$ are equal to or greater than zero you will run into a problem. You namely want to $x_5$, $x_6$ and $x_7$ also to be equal to or greater than zero, thus the optimal variable which you would like to increase from zero would need to have the largest positive gain for $x_3$ and the smallest negative gain for $x_4$. This is true for $x_5$, however even this variable will not be able to keep both $x_3$ and $x_4$ positive, namely for $x_5=\frac{1}{4}$ then $x_4=0$ but $x_3=-\frac{1}{4}$and for $x_5=\frac{1}{3}$ then $x_3=0$ but $x_4=-\frac{1}{18}$. Thus is can be concluded that there exists no solution such that $x_i\geq0$.
