If $x\left[ {\begin{array}{cccccccccccccccccccc}2\\3\end{array}} \right] + y\left[ {\begin{array}{cccccccccccccccccccc}{ - 1}\\1\end{array}} \right] = \left[ {\begin{array}{cccccccccccccccccccc}{10}\\5\end{array}} \right],$find the values of x and y.

.:

We have, $x\left[ {\begin{array}{cccccccccccccccccccc}2\\3\end{array}} \right] + y\left[ {\begin{array}{cccccccccccccccccccc}{ - 1}\\1\end{array}} \right] = \left[ {\begin{array}{cccccccccccccccccccc}{10}\\5\end{array}} \right],$

$\Rightarrow$ $\left[ {\begin{array}{cccccccccccccccccccc}{2x}\\{3x}\end{array}} \right] + \left[ {\begin{array}{cccccccccccccccccccc}{ - y}\\y\end{array}} \right] = \left[ {\begin{array}{cccccccccccccccccccc}{10}\\5\end{array}} \right] \Rightarrow \left[ {\begin{array}{cccccccccccccccccccc}{2x - y}\\{3x + y}\end{array}} \right] = \left[ {\begin{array}{cccccccccccccccccccc}{10}\\5\end{array}} \right]$

$\Rightarrow$ $2x - y = 10$ ….(i) and $3x + y = 5$ …..(ii)

Solving (i) \& (ii),

we get
x = 3 and y = $-$ 4

Hence, x = 3 and y =$-$ 4.