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: $$x_1 - x_2 = 400$$