I have three functions, but I cant get roots of these equations. In RegionPlot, what should I do, i can plot a vertical line through one of points？

Question

I have three functions, but I can't get roots for these equations. In RegionPlot, what should I do to plot a vertical line through one of the points? I can't get roots because there are implicit functions for more time, so I want to plot a vertical line through one of the roots, but I don't know how to write code to represent it.

f[x_, y_, z_] = x + y + z
g[x_, y_, z_] = x y z
h[x_, y_, z_] = x^7 + y^6 + z^5
Solve[{f[x, y, z] == g[x, y, z] == h[x, y, z]}, {x, y, z}]
    Manipulate[
 RegionPlot[
  f[x, y, z] > g[x, y, z] && f[x, y, z] > h[x, y, z], {x, 0, 5}, {y, 
   0, 5}], {z, -5, 5}]

Answer 1

We can solve f[x,y,z]-g[x,y,z]==0 && f[x,y,z]-h[x,y,z]==0 instead of f[x,y,z]==g[x,y,z]==h[x,y,z]

Clear[f, g, h];
f[x_, y_, z_] = x + y + z;
g[x_, y_, z_] = x y z;
h[x_, y_, z_] = x^7 + y^6 + z^5;
(*SolveValues[{f[x,y,z]-g[x,y,z]==0,f[x,y,z]-h[x,y,z]==0},{x,y},Reals]*)
Manipulate[
 Module[{pt}, 
  pt = NSolveValues[{f[x, y, z] - g[x, y, z] == 0, 
     f[x, y, z] - h[x, y, z] == 0}, {x, y}, Reals];
  RegionPlot[
   f[x, y, z] > g[x, y, z] && f[x, y, z] > h[x, y, z], {x, 0, 5}, {y, 
    0, 5}, Epilog -> {Green, Point[pt], Red, 
     InfiniteLine[pt[[-1]], {0, 1}]}, 
   PerformanceGoal -> "Quality"]], {z, -5, 5}]