Step $1$: Develop a system of linear equations for the network by writing an equation for each router $($A$,$ B$,$ C$,$ D$,$ and E$).$ Write the final answer as Ax$=$b$,$ where A is the $5$x$5$ coefficient matrix, x is the $5$x$1$ vector of unknowns, and b is a $5$x$1$ vector of constants.

To construct the coefficient matrix A and the constant vector b$,$ we need to write down the equations based on the network diagram. Each equation represents the flow of data into and out of each router. Based on the given information, the equations can be written as follows: