You are employed as a network engineer and have been asked to analyze a communication network to determine the current data rates and

ensure that the links aren't at risk of "reaching capacity." In the following figure of the network, the sender is transmitting data at a total rate of

$100+50=150$ megabits per second $($Mbps$).$ The data is transmitted from the sender to the receiver over a network of five different routers.

These routers are labeled A$,$ B$,$ C$,$ D$,$ and E$.$ The connections and data rates between the routers are labeled as $x_1,x_2,x_3,x_4,$ and $x_5.$

Develop a system of linear equations for the network by writing an equation for each router $($A$,$ B$,$ C$,$ D$,$ and E$).$ Make sure to write your

final answer as $Ax=b$ where $A$ is the $55$ coefficient matrix, $x$ is the $51$ vector of unknowns, and $b$ is a $51$ vector of constants.

Use MATLAB to construct the augmented matrix $Ab$ and then perform row reduction using the rref$()$ function. Write out your reduced

matrix and identify the free and basic variables of the system.

Use MATLAB to compute the LU decomposition of $A,$ i$.$e$.,$ find $A=LU.$ For this decomposition, find the transformed set of equations

$Ly=b.$ Solve the system of equations $Ly=b$ for the unknown vector $y.$