Question
Given $\left[\begin{array}{cc}2 & 1 \\ -3 & 4\end{array}\right] x=\left[\begin{array}{l}7 \\ 6\end{array}\right]$ Write the matrix $x$
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Answer
$ \begin{array}{l} \text { Let } x=\left[\begin{array}{l} x \\ y \end{array}\right]\end{array} $
$\therefore\left[\begin{array}{cc} 2 & 1 \\ -3 & 4 \end{array}\right] \times\left[\begin{array}{l} x \\ y \end{array}\right]=\left[\begin{array}{l} 7 \\ 6 \end{array}\right] $
$ \Rightarrow\left[\begin{array}{c} 2 x+y \\ -3 x+4 y \end{array}\right]=\left[\begin{array}{l} 7 \\ 6 \end{array}\right] \\ \Rightarrow 2 x+y=7 \ldots(1) \\ -3 x+4 y=6 \ldots(2) $
Multiplying by $4$ in equation $(1)$ and solving with equation $(2)$
$ \begin{array}{l} 8 x+4 y=28 \\ -3 x+4 y=6 \\ ( \pm) \_(z) \_(z) \\ 11 x=22 \\ x=2 \end{array} $
Putting the value of $x$ in equation $(1),$ we get
$ \begin{array}{l} \therefore 2 \times 2+y=7 \\ y=7-4=3 \end{array} $
$\therefore$ The matrix $x =\left[\begin{array}{l}x \\ y\end{array}\right]=\left[\begin{array}{l}2 \\ 3\end{array}\right]$
$\therefore\left[\begin{array}{cc} 2 & 1 \\ -3 & 4 \end{array}\right] \times\left[\begin{array}{l} x \\ y \end{array}\right]=\left[\begin{array}{l} 7 \\ 6 \end{array}\right] $
$ \Rightarrow\left[\begin{array}{c} 2 x+y \\ -3 x+4 y \end{array}\right]=\left[\begin{array}{l} 7 \\ 6 \end{array}\right] \\ \Rightarrow 2 x+y=7 \ldots(1) \\ -3 x+4 y=6 \ldots(2) $
Multiplying by $4$ in equation $(1)$ and solving with equation $(2)$
$ \begin{array}{l} 8 x+4 y=28 \\ -3 x+4 y=6 \\ ( \pm) \_(z) \_(z) \\ 11 x=22 \\ x=2 \end{array} $
Putting the value of $x$ in equation $(1),$ we get
$ \begin{array}{l} \therefore 2 \times 2+y=7 \\ y=7-4=3 \end{array} $
$\therefore$ The matrix $x =\left[\begin{array}{l}x \\ y\end{array}\right]=\left[\begin{array}{l}2 \\ 3\end{array}\right]$
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