First, write down all the equations at each node. We know that the incoming flow at each node will be equal to the outgoing flow.

For node A:

\({x_2} + 30 = {x_1} + 80\)

For node B:

\({x_3} + {x_5} = {x_2} + {x_4}\)

For node C:

\({x_6} + 100 = {x_5} + 40\)

For node D:

\({x_4} + 40 = {x_6} + 90\)

For node E:

\({x_1} + 60 = {x_3} + 20\)

The total flow into the system:

\(230 = 230\)

Re-arrange all the above equations to get the augmented matrix.

\(\begin{array}{c}{x_1} - {x_2} = - 50\\{x_2} - {x_3} + {x_4} - {x_5} = 0\\{x_5} - {x_6} = 60\\{x_4} - {x_6} = 50\\{x_1} - {x_3} = - 40\end{array}\)

Write down the equations in an augmented matrix form.

\(\left[ {\begin{array}{*{20}{c}}1&{ - 1}&0&0&0&0&{ - 50}\\0&1&{ - 1}&1&{ - 1}&0&0\\0&0&0&0&1&{ - 1}&{60}\\0&0&0&1&0&{ - 1}&{50}\\1&0&{ - 1}&0&0&0&{ - 40}\end{array}} \right]\)

Reduce the augmented matrix into an echelon matrix. Hence, the echelon matrix is:

\(\left[ {\begin{array}{*{20}{c}}1&0&{ - 1}&0&0&0&{ - 40}\\0&1&{ - 1}&0&0&0&{10}\\0&0&0&1&0&{ - 1}&{50}\\0&0&0&0&1&{ - 1}&{60}\\0&0&0&0&0&0&0\end{array}} \right]\)

a.Hence, the general solution of the traffic pattern of the network is \[\left\{ \begin{array}{l}{x_1} = {x_3} - 40\\{x_2} = {x_3} + 10\\{x_3}\,{\rm{is}}\,{\rm{free}}\\{x_4} = {x_6} + 50\\{x_5} = {x_6} + 60\\{x_6}\,{\rm{is}}\,{\rm{free}}\end{array} \right..\]

a.Hence, the general solution of the traffic pattern of the network is \(\left\{ \begin{array}{l}{x_1} = {x_3} - 40\\{x_2} = {x_3} + 10\\{x_3}\,{\rm{is}}\,{\rm{free}}\\{x_4} = {x_6} + 50\\{x_5} = {x_6} + 60\\{x_6}\,{\rm{is}}\,{\rm{free}}\end{array} \right..\)

b. To get the minimum values of the branches, \({x_1}\) cannot be negative as the value of \({x_3}\)will be greater than or equal to 40.

From equation\({x_2} = {x_3} + 10\) , if \({x_3}\)is greater than or equal to 40, then the value of \({x_2}\)will be equal to or greater than 50.

Since \({x_6}\) cannot be negative, values of \({x_4}\)and \({x_5}\)will be equal to or greater than 50 and 60, respectively.

Hence, the minimum flows of the branches \({x_2},{x_3},{x_4}\), and \({x_5}\) are \(50,40,50\), and \(60\), respectively.