Let $\overline{a},\overline{b}$ are non - collinear vectors if  $\overline{\alpha }=(x+4y)\overline{a}+(2x+y+1)\overline{b}$   $\overline{\beta }=(y-2x+2)\overline{a}+(2x-3y-1)\overline{b}$ are such that

$3\overline{\alpha }=2\overline{\beta }$  then find  $x, y.$

$3\alpha =2\beta \Rightarrow 3\left[\right(x+4y)a+(2x+y+1\left)b\right]=2$

$\left[\right(y-2x+2)a+(2x-3y-1\left)b\right]$

on comparing the coefficienents and b, we have 

$3x+12y=2y-4x+4$

$\Rightarrow 7x+10y=4\dots ...\left(1\right)$

and $6x+3y+3=4x-6y-2$

$\Rightarrow 2x+9y=-5\dots ...\left(2\right)$

solving $\left(1\right)$ and $\left(2\right)$

$x=2, y=-1$