a = {{2, -1, 1}, {1, 1, 2.}}; b = {13, 1};
{u, w, v} = SingularValueDecomposition[a];
u.w.Transpose[v]
f[x_] := If[x == 0, 0, 1/x]; SetAttributes[f, Listable];
w1 = f[Transpose[w]]
xx = v.w1.Transpose[u].b

Clear[x, y, z]; r = Thread[a.{x, y, z} == b]
d = 7; Show[
 ContourPlot3D[Evaluate[r], {x, -d, d}, {y, -d, d}, {z, -d, d}],
 Graphics3D[{Red, PointSize[0.05], Sphere[xx]}]]