Maximum principle for first-order quasilinear pde

Question

Let's suppose that $u\in C^{1}(K[0,1])$ solves $$a(x,y)u_{x}+b(x,y)u_{y}=-u$$ in $K[0,1]$. I need to show that if $$a(x,y)x+b(x,y)y>0$$ for all $(x,y)\in\partial K[0,1]$ then $u\equiv0$ in $K[0,1]$. One basic idea is to conclude that $min_{K[0,1]}u\geq0$ and $max_{K[0,1]}u\leq0$ but I don't recall any maximum principle for first-order quasilinear pde. Does anybody know how to attack this? Or should I try with method of characteristics?