Solution :

For the given sets of equations,

The Matrix form is given by

$\begin{bmatrix}2 & 3 & 1 \\ 5 & 1 & 1 \\ 3 & 2 & 4\end{bmatrix} \begin{bmatrix}x \\ y \\ z\end{bmatrix}=\begin{bmatrix}-1 \\ 9 \\ 11\end{bmatrix}$

As the form of A X=B

Let A be assumed to be L U

By comparing both sides

$ \begin{array}{l} a=2, b=5, d=3, g=\frac{3}{2}, c=\frac{-13}{2}, \\ e=\frac{-5}{2}, h=\frac{1}{2}, f=\frac{40}{13}, i=\frac{3}{13} \end{array} $

Now L Y=B where Y=U X

$ \left[\begin{array}{lll} 2 & 0 & 0 \\ 5 & \frac{-13}{2} & 0 \\ 3 & \frac{-5}{2} & \frac{40}{13} \end{array}\right]\left[\begin{array}{l} x \\ y \\ z \end{array}\right]=\left[\begin{array}{l} -1 \\ 9 \\ 11 \end{array}\right] $

Comparing Directly,

$\mathrm{y}_{1}=\frac{-1}{2}, \mathrm{y}_{2}=\frac{-23}{13}, \mathrm{y}_{3}=\frac{21}{8}$

Assume U X=Y

$\left[\begin{array}{III}1 & \frac{3}{2} & \frac{1}{2} \\ 0 & 1 & \frac{3}{13} \\ 0 & 0 & 1\end{array}\right] \left[\begin{array}{I} x \\ y \\ z\end{array}\right]=\left[\begin{array}{c}\frac{-1}{2} \\ \frac{-23}{12} \\ \frac{21}{8}\end{array}\right]$

Comparing both sides we get,

$\mathrm{z}=\frac{21}{8}$ Thus, the value of y is $\frac{21}{8}$.