Transformations

I don't really get 'some' transformation matrices. The ones that have a number other than 0 in $a_{12}$ or $a_{21}$ .

For instance:

Take $y=x^2$ . The following transformation matrix is applied,

$X'=T \cdot X$

$\iff \begin{bmatrix} x' \\ y' \end{bmatrix}=\begin{bmatrix} 1 & 2 \\ 2 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix}$

Actually, it's all fine and dandy when I do the matrix arithmetic ( $X=T^{-1}X'$ ), but I find it kinda hard to see what's going on. When the transformation is turned into equations, with (x,y) mapping onto (x',y'):

$x' = x+2y$

$y' = 2x+y$

Then how would I find the new equation without matrices? Thanks!