Answer
See answer
Work Step by Step
Intersection Flow in Flow out
\[
\begin{array}{lccc}
A & x_{1} & = & x_{3}+x_{4}+40 \\
B & 200 & = & x_{1}+x_{2} \\
C & x_{2}+x_{3} & = & x_{5}+100 \\
D & x_{4}+x_{5} & = & 60
\end{array}
\]
Total Flow $200=200$
Write the equations for each node:
\[
\begin{aligned}
x_{1} &-x_{3}-x_{4} & &=40 \\
x_{1}+x_{2} & & &=200 \\
& x_{2}+x_{3} & &-x_{5}=100 \\
& & x_{4}+x_{5} &=60
\end{aligned}
\]
Rearrange the equations:
\[
\left[\begin{array}{cccccc}
1 & 0 & -1 & -1 & 0 & 40 \\
1 & 1 & 0 & 0 & 0 & 200 \\
0 & 1 & 1 & 0 & -1 & 100 \\
0 & 0 & 0 & 1 & 1 & 60
\end{array}\right]
\]
Augmented Matrix
\begin{aligned}
&\left[\begin{array}{cccccc}
1 & 0 & -1 & -1 & 0 & 40 \\
0 & 1 & 1 & 1 & 0 & 160 \\
0 & 1 & 1 & 0 & -1 & 100 \\
0 & 0 & 0 & 1 & 1 & 60
\end{array}\right]\\
&R_{2}=R_{2}-R_{1}\\
&\left[\begin{array}{cccccc}
1 & 0 & -1 & -1 & 0 & 40 \\
0 & 1 & 1 & 1 & 0 & 160 \\
0 & 0 & 0 & -1 & -1 & -60 \\
0 & 0 & 0 & 1 & 1 & 60
\end{array}\right]\\
&R_{3}=R_{3}-R_{2}\\
&\left[\begin{array}{cccccc}
1 & 0 & -1 & -1 & 0 & 40 \\
0 & 1 & 1 & 1 & 0 & 160 \\
0 & 0 & 0 & -1 & -1 & -60 \\
0 & 0 & 0 & 0 & 0 & 0
\end{array}\right]\\
&R_{4}=R_{4}+R_{3}
\end{aligned}
\[
\left[\begin{array}{cccccc}
1 & 0 & -1 & -1 & 0 & 40 \\
0 & 1 & 1 & 1 & 0 & 160 \\
0 & 0 & 0 & 1 & 1 & 60 \\
0 & 0 & 0 & 0 & 0 & 0
\end{array}\right]
\]
The general solution
