$(25\%)$ Systems or Linear Equalions.

Consider the following system of linear equations:

$-x_2+4x_3+2x_4=30$

$18x_1+4x_2-6x_3-2x_4+2x_5=-68$

$-3x_1+9x_2+6x_3-3x_4+21x_5=-117$

$10x_2-2x_3+5x_4+2x_5=23$

$-x_1+3x_2-6x_4+2x_5=-23$

Problem $1\mathrm{.}1.(4\mathrm{.}5\%)$ Express the given system of linear equations as an augmented matrix $AM=(Alb)$ in MATLAB, where $A$ is the coefficient matrix and $b$ is the vector of constants. Then, perform partial pivoting in the augmented matrix AM using basic matrix operations $($i$.$e$.,$ by deleting and adding rows to the augmented matrix as needed$).$ Show each step of the partial pivoting process.

Problem $1\mathrm{.}2.(10\%)$ Use MATLAB and the result of problem $1\mathrm{.}1$ to solve the system of linear equations, considering the Gauss$-$Seidel method. Assume the initial solution as $x_1=10,x_2=-5,x_3=-5,x_4=-8,$ and $x_5=20.$ Perform iterations until the absolute approximate percentage error is below $2\%$ for all the unknowns of the system of linear equations.

Problem $1\mathrm{.}3.(4\%)$ Provide a plot that overlays the numerical values of all the unknowns of the system of linear equations versus the iterations conducted in the Gauss$-$Seidel method $($each unknown must be shown as a colored line$).$ Include axes labels, axes grid lines, and a legend that shows the unknowns and their respective colored lines.

Problem $1\mathrm{.}4.(4\%)$ Provide a plot that overlays the absolute approximate percentage errors of all the unknowns of the system of linear equations versus the iterations conducted in the Gauss$-$Seidel method $($each absolute approximate percentage error must be shown as a colored line$).$ Include axes labels, axes grid lines, and a legend that shows the absolute approximate percentage errors and their respective colored lines.

Problem $1\mathrm{.}5.(2\mathrm{.}5\%)$ Determine the absolute true percentage errors of the estimates obtained in problem $1\mathrm{.}2$ by first solving the system of linear equations using an analytical method $($e$.$g$.,$ Gauss elimination, Cramer's rule, or inverse of the coefficient matrix$).$ You are allowed to use MATLAB's built$-$in functions to solve the system of linear equations with an analytical method.