Answer
$x=\dfrac{u+2v}{5}$ and $y=\dfrac{3v-u}{10}$
Work Step by Step
The parallelogram with vertices $(0,0), (4,3), (2,4), (-2,1)$ is:
$-10 \lt 3x-4y \lt 0$; $0 \lt x+2y \lt 10$
Plug $u=3x-4y$ and $v=x+2y$
and $-10 \lt 3x-4y \lt 0$; $0 \lt x+2y \lt 10 \implies -10 \lt u \lt 0$ and $0 \lt v \lt 10$
This shows that the rectangle is in the uv plane.
Further, we have $3x-4y+2x+4y=u+2v$
$\implies x=\dfrac{u+2v}{5}$
and t $3x-4y-3x-6y=u-3v \implies y=\dfrac{3v-u}{10}$
Hence, $x=\dfrac{u+2v}{5}$ and $y=\dfrac{3v-u}{10}$