Question

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

Answer 1

Given : $x\left[\begin{array}{c}2\\ 3\end{array}\right]+y\left[\begin{array}{c}-1\\ 1\end{array}\right]=\left[\begin{array}{c}10\\ 5\end{array}\right]$

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

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

Comparing
$2x-y=10\cdots \left(i\right)$
$3x+y=5\cdots \left(ii\right)$
Adding $\left(i\right)$ and $\left(ii\right)$
$\Rightarrow (2x-y)+(3x+y)=10+5$
$\Rightarrow 2x-y+3x+y=15$
$\Rightarrow 2x+3x-y+y=15$
$\Rightarrow 5x+0=15$
$\Rightarrow x=\frac{15}{5}$
$\Rightarrow x=3$
Putting value of $x$ in $2x-y=10$
$\Rightarrow 2\left(3\right)-y=10$
$\Rightarrow 6-y=10$
$\Rightarrow -y=10-6$
$\Rightarrow y=-4$
Therefore, $x=3$ and $y=-4$.