Not doing things

안 is a clipped form of 아니, the same root you met in 아니요 and 아니에요 on day 5. It goes in front because the end of a Korean sentence is already crowded: tense, politeness and whether it is a question all live in the ending. Putting the negative before the verb leaves that machinery untouched, so 안 먹어요, 안 먹었어요 and 안 먹어요? need no new endings at all. The rule to hold on to is that 안 touches the verb, so anything else, the object or a time word, goes before 안.

-지 않아요 is built from the verb stem, the linking syllable 지, and 않다, which is itself 아니 plus 하다, roughly 'not doing'. Because 않 has ㅏ as its vowel, it takes 아요 by the rule from day 13, which is why you hear 않아요. The long form is common in writing, news and careful speech; in conversation 안 is far more frequent, and using the long form every time sounds a little stiff. With 하다 verbs, 안 must sit right against 해요, because 안 negates a verb and 공부 on its own is a noun.

The choice between 안 and 못 is a choice between will and ability. 안 가요 reports a decision, so to the person who invited you it can sound like 'I do not want to'. 못 가요 says circumstances decided for you, which is why Koreans reach for 못 when turning something down, even when they could, at a push, go. Both sit in the same place before the verb and split 하다 verbs in the same way, so once you can place one you can place the other.