Direction Field

Brannon and Boyce 3.2.19

df=VectorPlot[{x+y-3,-x+y+1}/Norm[{x+y-3,-x+y+1}],{x,0,4},{y,0,2}]

soln[xzero_?NumericQ,yzero_?NumericQ]:=NDSolve[{dx'[t]==dx[t]+dy[t]-3,dy'[t]==-dx[t]+dy[t]+1,dx[0]==xzero,dy[0]==yzero},{dx,dy},{t,-10,10}]

xx[t_?NumericQ,xzero_?NumericQ,yzero_?NumericQ]:=dx[t]/.soln[xzero,yzero][[1]]
yy[t_?NumericQ,xzero_?NumericQ,yzero_?NumericQ]:=dy[t]/.soln[xzero,yzero][[1]]

xx[-1,3.5,0.1]

orbit1=ParametricPlot[{xx[s,3.5,0.1],yy[s,3.5,0.1]},{s,-2,0},PlotStyle->{Thick,Red}];

Show[df,orbit1]

The equilibrium solves x+y = 3 and x-y = 1.  That is (x*,y*) = (2,1).