Mathematics-I (3110014)

$\mathrm{Here},$

$-\mathrm{x}+3\mathrm{y}+4\mathrm{z} =30$
$3\mathrm{x}+2\mathrm{y}-\mathrm{z} =9$
$2\mathrm{x}-\mathrm{y}+2\mathrm{z} =10$

$\mathrm{co}-\mathrm{efficient} \mathrm{matrix} \mathrm{A}=\left[\begin{array}{ccc}-1& 3& 4\\ 3& 2& -1\\ 2& -1& 2\end{array}\right]$

$\mathrm{Variable} \mathrm{matrix} \mathrm{X}=\left[\begin{array}{c}\mathrm{x}\\ \mathrm{y}\\ \mathrm{z}\end{array}\right]$

$\mathrm{Constant} \mathrm{matrix} \mathrm{B}=\left[\begin{array}{c}30\\ 9\\ 10\end{array}\right]$

$\mathrm{Now}, \mathrm{Augmented} \mathrm{Matrix} \mathrm{is},$

$\left[\overline{)\mathrm{A} } \mathrm{B}\right]=\left[\overline{)\begin{array}{ccc}-1& 3& 4\\ 3& 2& -1\\ 2& -1& 2\end{array}} \begin{array}{c}30\\ 9\\ 10\end{array}\right]$

$\mathrm{Taking} {\mathrm{R}}_1(-1)$

$\left[\overline{)\mathrm{A} } \mathrm{B}\right]~\left[\overline{)\begin{array}{ccc}1& -3& -4\\ 3& 2& -1\\ 2& -1& 2\end{array}} \begin{array}{c}-30\\ 9\\ 10\end{array}\right]$

$\mathrm{Taking} {\mathrm{R}}_{12}(-3) , {\mathrm{R}}_{13}(-2)$

$\left[\overline{)\mathrm{A} } \mathrm{B}\right]~\left[\overline{)\begin{array}{ccc}1& -3& -4\\ 3-3\left(1\right)& 2-3(-3)& -1-3(-4)\\ 2-2\left(1\right)& -1-2(-3)& 2-2(-4)\end{array}} \begin{array}{c}-30\\ 9-3(-30)\\ 10-2(-30)\end{array}\right]$

$\left[\overline{)\mathrm{A} } \mathrm{B}\right]~\left[\overline{)\begin{array}{ccc}1& -3& -4\\ 0& 11& 11\\ 0& 5& 10\end{array}} \begin{array}{c}-30\\ 99\\ 70\end{array}\right]$

$\mathrm{Taking} {\mathrm{R}}_2\left(\frac{1}{11}\right) , {\mathrm{R}}_3\left(\frac{1}{5}\right)$

$\left[\overline{)\mathrm{A} } \mathrm{B}\right]~\left[\overline{)\begin{array}{ccc}1& -3& -4\\ 0& 1& 1\\ 0& 1& 2\end{array}} \begin{array}{c}-30\\ 9\\ 14\end{array}\right]$

$\mathrm{Taking} {\mathrm{R}}_{32}\left(-1\right)$

$\left[\overline{)\mathrm{A} } \mathrm{B}\right]~\left[\overline{)\begin{array}{ccc}1& -3& -4\\ 0& 1& 1\\ 0-1\left(0\right)& 1-1\left(1\right)& 2-1\left(1\right)\end{array}} \begin{array}{c}-30\\ 9\\ 14-1\left(9\right)\end{array}\right]$

$\left[\overline{)\mathrm{A} } \mathrm{B}\right]~\left[\overline{)\begin{array}{ccc}1& -3& -4\\ 0& 1& 1\\ 0& 0& 1\end{array}} \begin{array}{c}-30\\ 9\\ 5\end{array}\right]$

$\mathrm{By} \mathrm{back} \mathrm{substitution},$

• $\mathrm{z}=5$

• $\mathrm{y}+\mathrm{z}=9$
$⟹\mathrm{y}=9-\mathrm{z}$
$⟹\mathrm{y}=9-5$
$⟹\mathrm{y}=4$

• $\mathrm{x}-3\mathrm{y}-4\mathrm{z}=-30$
$⟹\mathrm{x}=-30+3\mathrm{y}+4\mathrm{z}$
$⟹\mathrm{x}=-30+3\left(4\right)+4\left(5\right)$
$⟹\mathrm{x}=-30+12+20$
$⟹\mathrm{x}=2$

$\mathrm{Hence}, \mathrm{solution} \mathrm{is} (\mathrm{x},\mathrm{y},\mathrm{z})=(2,4,5).$