# Linear Algebra: Network Flow problem

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.

• The total coming into any particular node must equal the total coming out. On the top left node, you have $400+x_2$ coming in, and $x_1$ coming out, so the equation should be $x_1 = 400+x_2$, or $x_1-x_2=400$; you've got the left hand side multiplied by $-1$. – Arturo Magidin May 23 '11 at 18:14
• Thanks for the explanation. – user7814 May 23 '11 at 18:21

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