I can only speak in broad terms - ie, in the equation - because my knowledge is limited to the doco I watched last night, and I dont know any of the equations. Im counting on you guys to know them.
But the point is that it's not about equations at all. Dark matter is, at its most fundamental, simply an observation. The equations only come later when we try to figure out what could be causing that observation. Or in this case, as already mentioned, a whole set of independent observations.
I believe the earth has an iron core, but I understand its represented in different ways. We dont agree on its specifics. Dark matter strikes me as the same thing - way into the grey area.
That's actually a pretty good example. To start with, we make a variety of observations. First we notice that the Earth has a global magnetic field. Then, completely separately, we look at the way different waves from Earthquakes spread across the planet. Both these observation, and others, lead us to the conclusion that there is a region inside the planet with properties different from the rocks we can actually see, and we decide to call it the core. We then start trying to figure out from all these observations exactly what the core is and how and why it results in these particular observations and not something else.
It's exactly the same for dark matter. We have a set of observations, including things such as the details of how things orbit within galaxies, what the cosmic background radiation looks like, how fast galaxies themselves are moving, and so on. All these separate observations taken together lead us to the conclusion that there is some gravitational influence that we hadn't previously noticed. We call that dark matter. Just as with the Earth's core, that's nothing more than a label for something that starts off being almost entirely unknown. And just like the core, we then use all those observations to try to work out exactly what it is and how it works.
Ultimately, there's nothing special about dark matter. The reasoning behind it is exactly the same as the reasoning behind every scientific discovery - notice something that doesn't quite fit with existing ideas, give it a name and try to figure out what's going on. The only real difference is the difficulty in first discovering and then later investigating it. And the reason for that is quite simple - we've already discovered and investigated the easy stuff. If you can see and pick up something with your human eyes and hands, then so could people thousands of years ago. So most (I won't say all since there can always be stuff we've missed) of the things left to discover are things that unassisted human senses can't detect themselves. That doesn't make them any less real, and it doesn't mean it's all just silly people playing with meaningless equations, but it does mean it can be hard for the layperson to accept since there's just no way to understand them without first understanding all the equations and complex equipment needed to detect them.
