"I am not busy."
Adjectives take 不 or 沒 to become negative, if allowed to. 不是 is not acceptable (at least not in a plain statement).
First of all, chinese has two copulas (to be): one for adjectives, and one for else. The verb to be with adjectives is 很 hèén (my keybord doesnt have funny accents :/ ). Here you can figure what verb it is by context, as chinese is very context sensitive.
很 looks like a copula but I've asked around and it is not a copula but an adverb literally meaning "very" but usually without meaning when used in these kind of constructions.
Many languages don't use copulas with adjectives since their adjectives are more like verbs than the adjectives in European languages are. They are often called "stative verbs".
I also just learned in the comments of another question in this lesson that 很 is not used in questions or negative statements.
I believe that if you used 很 in a question or negative statement it would really mean "very" (not be a grammatical filler word). But in these cases it may be more common to use other intensifiers such as 非常.
是 is only used when the sentence is "subject" + "verb" (是) + "object", like in:
我是老师 ("I'm a teacher).
In this exercise the sentence is: "subject" + "verb" + "adjective". And the reason there is no 很 either, was explained already by another user above.
Why do other similar questions require you to write 很忙 while this one is only 忙?
When you make an absolute negation in Chinese, it is not correct to talk about the magnitude, quantity, completion, etc. As I am not busy (at all), "busy" is inexistent and I cannot attach 很 and say 我很不忙 ↢wrong sentence
(The exceptions I can think of are some feelings and emotions. It is allowed to say 我很不开心/I am very unhappy, 我很不喜欢～/I don't like ~ very much.)
If I say 我不很忙，I am making a negation on "very", not busy. It means I am still busy, just not very busy.
You seem to be saying that 很 always means "very" whereas Chinese language courses generally teach us very early that when used before an adjective it does not carry that meaning. Which is true?
I think in a thread another native user pointed out to you 很 does always mean "very". I agree with her.
I know a theory says it's a copula. I don't study linguistics, not even language academically. So I would hold back saying it's wrong. It didn't convince me anyway. If it is a copula, 2 questions need to be answered: (1) Why it disappears in so many sentence patterns? (2) Why it can always be replaced with a phrase of magnitude?
I am okay for courses to tell people to always insert a 很 - sometimes it is an inevitable evil to make things simpler for learning. However they should teach people it is a kind of mnemonics, not as a "rule", before they can explain things well.
I don't think it's a copula. What I had learned was that it is neither a copula nor always a literal "very" but something unique to Chinese.
For instance, how would you translate this sentence into Chinese?
"I am tall but you are very tall."
There would be no literal translation you guys keep looking for. You always need to ensure the degree of magnitude is conceived explicitly or via the context for adjectives like this. So it depends on the circumstances in which you would say this sentence. Can you describe a scenario in details you would say it? Are you trying to highlight a comparison, or are you providing 2 parallel descriptions? Does that "you" know how tall that "I" is? And does that "you" know how tall he is himself?
我不忙了 does not have the same meaning as 我不忙. The particle 了 indicates that you were previously just doing something that made you busy, but you are no longer busy. If you leave out the 了 and just have the sentence 我不忙, it is just a general statement saying 'I'm not busy'.