Network Flow Problem - How to solve with a matrix? Find the general flow pattern of the network. Assuming that the flows are all nonnegative, what is the smallest possible value for $x_4$?

Points of Intersection: Flow In = Flow Out

A: $x_1+x_4 = x_2$

B: $x_2 = x_3 + 100$

C: $x_3 + 80 = x_4$

The total flow in equals the total flow out, so you can set up other equations...

How do I turn this into a solvable matrix to get the answer?
Put it each column the corresponding variable, last column is used for constants. So rewrite each equation in the form $...=0$. Then, for $A$ we get $x_1-x_2+x_4=0$, $B$: $x_2-x_3-100=0$, $C$: $x_3-x_4+80=0$. Put that in a matrix:
$
 \begin{pmatrix}
1 &amp; -1 &amp; 0 &amp; 1 &amp; 0\\
0 &amp; 1 &amp; -1 &amp; 0 &amp; -100\\
0 &amp; 0 &amp; 1 &amp; -1 &amp; 80\\
 \end{pmatrix}
$

Solving that matrix results into $
 \begin{pmatrix}
1 &amp; 0 &amp; 0 &amp; 0 &amp; 20\\
0 &amp; 1 &amp; 0 &amp; -1 &amp; -20\\
0 &amp; 0 &amp; 1 &amp; -1 &amp; 80\\
 \end{pmatrix}
$

So, in other words, $x_1=20, x_2=-x_4-20, x_3=-x_4+80$. Since the flows are non-negative, it must hold that $x_4\le -20$ (for $x_2\ge 0$) and $x_4 \le 80$ for $x_3 \ge 0$. So, $x_4 \le -20$. But that is not possible, since $x_4 \ge 0$ so there are no solutions.
Sol'n: x1=20, x2=100, x3=0, x4=80
Coefficient matrix
[1 -1  0  1][x1]=[0]
[0  1 -1  0][x2]=[100]
[0  0  1 -1][x3]=[-80]
            [X4]