Determining the formula for a linear map Determine the formula for the following linear map:

$L : \mathbb{R}^2 \rightarrow \mathbb{R}^2$ with $L(1,2) = (0,-1)$ and $L(-1,-1) = (2,1)$.

Attempt at solution: On the basis of these examples I was able to guess the second element: everytime we have that $L(x,y) = (... , -x)$. But this method seems a bit random to me, and I don't know how to compute the first element. Is there some strategy to deal with these kind of problems? Maybe setting up some linear system? Help would be appreciated.
$$L(x,y)=(ax+by,cx+dy)$$
$$L(1,2)=(a+2b,c+2d)=(0,-1)$$
$$L(-1,-1)=(-a-b,-c-d)=(2,1)$$
This becomes two linear systems with two equations, yielding the solution $(a,b,c,d)=(-4,2,-1,0)$. That is, $L(x,y)=(-4x+2y,-x)$.
Hint: $(0, 1) = (1, 2) + (-1, -1)$, so $T(0, 1) = (0, -1) + (2, 1) = (2, 0)$. What is $T(1, 0)$?