a$)$ A software developer is in the process of creating a mobile app. The app's various aspects are represented by three variables:

$x$ : Represents the computational resources or processing power needed by the app.

$y$ : Denotes the storage requirements or memory usage of the app.

z: Indicates the data transfer demands or network bandwidth required by the app.

While developing the mobile app, the developer strives to meet certain criteria as per the following equations. First, a performance goal, such as the app's desired efficiency level or response time can be represented as follows:

$93x+87y+79z=48$

Second, a resource usage limit$,$ like the maximum permissible CPU utilization or memory consumption can be signified as follows:

$13x+29y+37z=19$

Last, a quality benchmark, possibly indicating the minimum user satisfaction or reliability expected from the app can be reflected as follows:

$47x+53y+61z=33$

By employing Gauss$-$Jordan Elimination to solve the above system of equations, the developer can determine the optimal values for processing power $(x),$ memory usage $(y),$ and network bandwidth $(z).$ This allows the developer to fine$-$tune the app's design or implementation to optimize performance, and resource usage, and meet quality standards.

Note: Please use fractions in your calculations.

$(10$ marks$)$

The matrix of a linear system has the form,

$\left[\begin{array}{ccc}-1& 2& 1\\ 1& 0& -1\\ 2& 1& 2\end{array}\right]\left[\begin{array}{c}{x}_1\\ x_2\\ x_3\end{array}\right]=\left[\begin{array}{c}3\\ 1\\ 2\end{array}\right]$

Solve the linear system using LU factorization using Crout method.