Learn Arabic (Modern Standard): هل أنت مشغول؟ أجل، أنا مشغول.

Not exactly. هل is a special question particle that has no direct English translation. You put it at the start of a sentence to turn it into a yes/no question. Because Arabic doesn't use words like "am" or "are" in the present tense, هل just signals that the statement "you busy" is being asked as a question.

I usually see نعم for yes. Why does this sentence use أجل?

Both نعم and أجل mean yes in Modern Standard Arabic. نعم is the standard, most common word for yes. أجل means yes but has a slightly formal feel, often used to affirm or agree with a factual statement, much like saying "Yes, indeed." You can safely use either one here.

Where are the words for "are" and "am"?

In Arabic, the present tense of the verb "to be" is simply left out. The phrase أنت مشغول literally translates to "you busy", and أنا مشغول translates to "I busy". The am, is, and are are naturally understood by the listener.

Could I use the other question particle, أ, instead of هل here?

Yes, you could say أأنت مشغول؟ to mean the exact same thing. However, هل is much more common before pronouns like أنت because putting two "a" sounds next to each other (the question particle أ and the start of أنت) can feel a bit clunky to say out loud.