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If $x\left[\begin{array}{l}2 \\ 3\end{array}\right]+y\left[\begin{array}{c}-1 \\ 1\end{array}\right]=\left[\begin{array}{c}10 \\ 5\end{array}\right]$ then find the values of $x$ and $y$.
Solution:
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Verified Answer
$x\left[\begin{array}{l}2 \\ 3\end{array}\right]+y\left[\begin{array}{c}-1 \\ 1\end{array}\right]=\left[\begin{array}{c}10 \\ 5\end{array}\right]$
$\Rightarrow\left[\begin{array}{l}2 x-y \\ 3 x+y\end{array}\right]=\left[\begin{array}{c}10 \\ 5\end{array}\right]$
$\Rightarrow \begin{aligned}&2 x-y=10 \\&3 x+y=5\end{aligned}$
...................
$5 x=15 \quad$ on addition,
$\begin{gathered}
x=3 \\
\therefore \quad 3 \times 3+y=5 \\
\Rightarrow y=-4
\end{gathered}$
Hence, $x=3$ and $y=-4$
$\Rightarrow\left[\begin{array}{l}2 x-y \\ 3 x+y\end{array}\right]=\left[\begin{array}{c}10 \\ 5\end{array}\right]$
$\Rightarrow \begin{aligned}&2 x-y=10 \\&3 x+y=5\end{aligned}$
...................
$5 x=15 \quad$ on addition,
$\begin{gathered}
x=3 \\
\therefore \quad 3 \times 3+y=5 \\
\Rightarrow y=-4
\end{gathered}$
Hence, $x=3$ and $y=-4$
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