First of all, that isn't correct, it just becomes trickier, because objects that are not point masses can rotate.(Original post by Oblogog)
It confuses me that the forces are acting at different points. My understanding was that you could only resolve forces acting at the same point.
You're correct that it doesn't make sense trying to resolve linear forces in general when applied at different points on an object, because they will each induce a rotation, due to being offset from the centre of mass
IF you are told the object is in equilibrium, that not only means that it is not accelerating, but that it is also not rotating, so forces acting in one direction will continue to act in that direction at all subsequent times, so you don't need to worry about rotation when summing the forces to find the equilibrium. You're using the fact that the rod doesn't rotate to simplify it as if you were talking about a point particle. After you've done that, you can again think about the moments of the forces, but as long as it's in equilibrium, directionally resolving the forces is valid.
Think of a flat table. You put an object on it, and the support force comes from the legs. Each leg is applying a force that is offset from the centre of mass, so has the potential to rotate the table ( remove three legs, and the table falls over) but because it is standing in equilibrium, you can say that the weight of the object is the same as the sum of the support forces from the legs. It's the exact same problem, just in 2D
(To anyone else reading, I am aware of the areas where I have made simplifications)
But we haven't even met!