You said: "the liberty to come together to work on a solution, via government."

I think this statement could be divided into two parts: coming together to work on a solution (implies voluntary action / non-coercion) and "via goverment" (implies non-voluntary action by agents of the state, and coercion).

Libertarians definitely support the liberty of people to come together (freedom of association). But we reject the broad use of governments to provide solutions (aside from some narrow areas: justice, police force, border defence).

When you mix poison with food, you don't get a happy medium, you get poisoned. If you mix state coercion with freedom of association, you simply get the tyranny of the majority.

Why can't people who believe in a particular solution (eg, communism, or Christian conservatism under a religious leader) simply apply that solution themselves rather than forcing everyone else to also follow it? For example: by living in a commune, Christian fellowship or by living their beliefs in their own lives.

(And yes I know that it isn't easy to do these things under the current system of laws, taxes and regulations - but it would be much easier in a libertarian system).

"via goverment" (implies non-voluntary action by agents of the state, and coercion)

I think this implies what you say it implies, because a Libertarian is reading it! Thus,

If you mix state coercion with freedom of association, you simply get the tyranny of the majority.

you take a very limited view of what can be accomplished via government. I'm also very critical of the notion of "majority rule" as it is often applied by statists; however, I'm still open to the idea of "a more perfect union." A signal fault of some Libertarian thinking seems to be in believing that "the state" and "the people" must always be entirely separate entities and often at odds -- true, this appears to be the case much of the time; but I am not convinced that it must always be the case. The more Libertarians argue from that line of thought, the more they would attempt to disenfranchise citizens of a nation, or convince the citizenry not to act via their effects upon government and their power to shape it -- so it seems like a 4GW mode of argument: "Just give up; it's useless!"

Another fault of much Libertarian thinking: While founding a moralistic argument upon "human liberty", Libertarians appear to assume that most humans are really incapable of, and unwilling to, form ideas and act upon them to ensure their own liberty. By this, I mean that governments did not spring from nowhere outside the consent of the governed. The reason we have governments and bureaucracies and so forth, is that people came together to form them. If government is such a poison (your metaphor), then many millions of individuals acting upon their own convictions have made that poison; thus the Libertarian seems to say, those many millions cannot be trusted to make such decisions about how they will come together. So,

Why can't people who believe in a particular solution (eg, communism, or Christian conservatism under a religious leader) simply apply that solution themselves rather than forcing everyone else to also follow it? For example: by living in a commune, Christian fellowship or by living their beliefs in their own lives.

--what is missed here is the fact that people have come together to form states. If you want a negative example, look at certain religious enclaves within America, particularly certain sects of fundamentalist Mormons: They come together to live as they think best, despite the opposition from general law; but many of those born into those enclaves -- say, young girls -- may be forced to marry while young, in arranged marriages to older members of the enclave. Or look at another example in American history: the slave-holding South, which wanted the freedom to live as they wish without outside interference. The communes of which you speak are merely smaller states.