How to join the lhs and rhs of the rows of a matrix equation and solve for one of the variables?

The matrix equation m.{x,y}==b results in an awkward disjointed looking presentation but correct construction of a system of linear equations {x + y, -x + y} == {1, 2}.

• How do I put the results of m.{x,y}==b in a more proper form such
  as eqs={x+y==1,-x+y==2}.
• How to solve eqs={x+y==1,-x+y==2} for y in terms of x so that
  the outcome looks like this: eqs={y==1-x,y==2+x}?

Here is the working code…

ClearAll[m,b,vars,x,y,eqs];
m = {{1, 1}, {-1, 1}};
b = {1, 2};
vars={x,y};
eqs = m.vars == b;