Thank you for your reply. Unfortunately, I get a series of error messages with your script. One is: Tag In is protected. Another is: {rule} is neither a list of replacement rules nor a valid dispatch \ table, and so cannot be used for replacing. >> My version of Mathematica is 10. Could it be incompatible with yours?

if your goal is to get the correct equation this code will generate it. However, plotting an ellipsoid can be accomplished much simpler.

In[]:= (*some parameters*)
rule = {a -> 0.2, b -> 1, c -> 2, rx -> 10. Degree, ry -> 15. Degree, 
   rz -> -30. Degree, tx -> -1, ty -> 1.5, tz -> 0.8};

(*3D equation for ellipsoid*)
eq = x^2/a + y^2/b + z^2/c == 1.;
(*apply rotation and translation to the x,y,z coordinates*)
rot = Thread[{x, y, 
     z} -> (RotationMatrix[rz, {0, 0, 1}] . 
      RotationMatrix[ry, {0, 1, 0}] . 
      RotationMatrix[rx, {1, 0, 0}] . ({x, y, z} - {tx, ty, tz}))];
(*equation for rotated and translated ellipsoid*)
eqRot = eq /. rot;

(*solve for z*)
eqPlot = z /. NSolve[eqRot, z];

(*fill in some numbers*)
plot = eqPlot /. rule
p1 = Plot3D[plot, {x, -3, 3}, {y, -3, 3}, 
   PlotRange -> {{-3, 3}, {-3, 3}, {-3, 3}}, BoxRatios -> 1, 
   ViewPoint -> Front];

(*using Ellipsoid*)
ellipsoid = 
  Translate[
   Rotate[Rotate[
     Rotate[Ellipsoid[{0, 0, 0}, DiagonalMatrix[{a, b, c}]], -rx, {1, 
       0, 0}], -ry, {0, 1, 0}], -rz, {0, 0, 1}], {tx, ty, tz}];
p2 = Graphics3D[(ellipsoid) /. rule, 
   PlotRange -> {{-3, 3}, {-3, 3}, {-3, 3}}, BoxRatios -> 1, 
   Axes -> True, ViewPoint -> Front];

GraphicsRow[{p1, p2, Show[p1, p2]}]

Out[]= {0.807372 (0.462989 - 1.14238 x - 0.409664 y - 
    1. Sqrt[(-0.462989 + 1.14238 x + 0.409664 y)^2 - 
      2.47717 (2.10003 + 1.21852 x + 3.76554 x^2 - 3.0741 y + 
         3.59912 x y + 2.11516 y^2)]), 
 0.807372 (0.462989 - 1.14238 x - 0.409664 y + 
    Sqrt[(-0.462989 + 1.14238 x + 0.409664 y)^2 - 
     2.47717 (2.10003 + 1.21852 x + 3.76554 x^2 - 3.0741 y + 
        3.59912 x y + 2.11516 y^2)])}