m = 4;
x = {1, 2, 3, 4};
y = {5, 8, 10, 15};
pod = 
        Table[{x[[i]], y[[i]]}, {i, m}];
sl1 = 
        ListPlot[pod, PlotRange -> {{0, 5}, {0, 20}}, 
          DisplayFunction ->  
            Identity];
  
G[k_, l_] = 
        Sum[((k/x[[i]]) + l - (1/y[[i]]))^2, {i, 
            m}];
   D[G[k, l], 
        k];
   D[G[k, l], 
        l];
rj = 
        Solve[ {D[G[k, l], k] == 0, D[G[k, l], l] == 0}, {k, 
              l}] // N;
k0 = k /. rj[[1]];
l0 = l /. rj[[1]];
a0 = 1/l0;
b0 = k0 a0;
sl2 = 
        Plot[a0 t/(b0 + t), {t, 0, 20}, PlotStyle ->  Hue[0.6], 
          DisplayFunction ->  
            Identity];
  
F[a_, b_] = 
        Sum[(a x[[i]]/(b + x[[i]]) - y[[i]])^2, {i, 
            m}];
rj2 = FindMinimum[F[a, b], {a, a0}, {b, b0}]
aopt = a /. rj2[[2]];
bopt = b /. rj2[[2]];
sl3 = 
        Plot[aopt t/(bopt + t), {t, 0, 20}, PlotStyle ->  Hue[1], 
          DisplayFunction ->  Identity];
Show[sl1, sl2, sl3, 
      DisplayFunction -> $DisplayFunction]