1 Attachment(s)

Traffic Flow, Network Analysis

Ok, I am having trouble completing the following problem:

The flow of traffic (in vehicles per hour) through a network of streets is shown in Figure 1.16 (my attachment).

(a) Solve this system for $\displaystyle x_i, i = 1, 2, ..., 5.$

(b) Find the traffic flow when $\displaystyle x_2 = 200 and x_3 = 50.$

(c) Find the traffic flow when $\displaystyle x_2 = 150 and x_3 = 0.$

Ok, I am having trouble with part a. I feel once I get part a, parts b and c should be a fairly straightforward, plug in the values and go from there. Using my figure I get the following Input = Output equations for the labeled junctions in my figure:

$\displaystyle A:x_1 + x_2 = 300$

$\displaystyle B:x_1 + x_3 - x_4 = 150$

$\displaystyle C:x_2 - x_3 - x_5 = -200$

$\displaystyle D:x_4 + x_5 = 350$

Anyhow, when I put it into matrix form and proceed to RREF I get:

$\displaystyle

\left(\begin{array}{cccccc}1&0&1&0&1&500\\0&1&-1&0&-1&-200\\0&0&0&1&1&350\\0&0&0&0&0&0\end{array}\right)

$

Because of the zero row I know $\displaystyle x_4$ and $\displaystyle x_5$ are free variables, so I set $\displaystyle x_4 = s$ and $\displaystyle x_5 = t $. It's at this point I get stuck. I have tried to solve it, but I wind up getting an expression for $\displaystyle x_2$ in both $\displaystyle x_1$ and $\displaystyle x_3$. Do I need to let another variable be a free variable? Thanks for any and all help.