Clear[x,y,z]
A={2,1,-1};
B={1,-3,-1};
c={x,y,z};
Solve[Dot[A,c]==2]

Cross[A,{x,y,z}/.z->-2+2 x+y]

Solve[{-2+2 x+2 y,4-5 x-2 y,-x+2 y}==B,{x,y}]