Fractions in multi-step equations

Having a bit of trouble wrapping my head around the following, as I am studying on my own, after years away from school. Got the basic concept down, but was hoping someone could explain how and WHY this works in different words so I can hopefully get a firmer grasp on it.

The first equation I have figured out pretty well, but I will explain it the way I understand it, and see if this is correct.

3/4x-6=10

Add six to both sides and you have:

3/4x=16

Now, when you multiply both sides by the reciprocal, the faction and the reciprocal will cancel each other out on the left side? Is that correct? Meaning, if 3/4 and 4/3 cancel each other out, you are left with

x=4/3*16/1?

If that is correct up until there, then I think I have a firm grasp on it. Someone explaining it in other words would be a help however.

Now, the second equation I am having a bit more trouble with. It involves two factions, where the variable is the numerator in one of the fractions.

x-8 = x/3+1/6

Now, apparently I want to get rid of the fractions. So I would multiply both sides by the least common multiple of the denominators, coming up with the following?

6(x-8) = 6(x/3+1/6)

Is that correct up until there? If it is, can someone explain to me why this works? Assuming this is correct, it would leave me with:

6x-48 = 2x+1

Then, to finish up, isolate 4x on the left side, isolate 49 on the left side, and end up with an answer of x = 49/4

I'm pretty sure this second equation is figured out correctly. I'm just having trouble grasping the concept of WHY multiplying by the least common multiple of all the denominators works.

Any help you guys can provide would be awesome. It'd be even awesome-er if someone could provide me with a few problems to work out, seeing as I'm studying on my own, and don't have a ready source of problems like the second one to work with.