[Math] Linear Algebra: Network Flow problem

Question

So I have the following problem:

And I have obtained the following system of equations:
$$\begin{align*}
-x_1+x_2&=400\\
x_1+x_3-x_4&=600\\
x_2+x_3+x_5&=300\\
x_4+x_5&=100
\end{align*}$$

By using the Gaussian elimination method, I found the following system of equations (using rref() on a TI83):

$$\begin{align*}
x_1+x_3+x_5&=0\\
x_2+x_3+x_5&=0\\
x_4+x_5&=1\\
0&=1
\end{align*}$$

I would like some clarification as to what I am doing wrong.

Thank you.

Answer 1

Your first balanced flow equation is incorrect, it should be: $$x_1 - x_2 = 400$$