I do homework while eating
I'm doing homework while eating
Is the second sentence correct? Can I use 在 or does 一边...一边 already imply that the action I'm talking about is happening right now?
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[一边 (verb A) 一边 (verb B)] indicates (verb A) and (verb B) are carried out concurrently. Please read Is the 边～边 the synonym for 一边～一边? for reference
[在 (verb A)] indicates (verb A) is in a progressive tense (in the middle of doing it / in a continuous state of doing it)
Neither structure indicate tense, because Chinese verbs do not inflect between present and past tense. Please read How do you specify past tense for 是? for reference
It is not technically wrong, but the more efficient structure to express the same meaning is "我在边吃饭，边做作业"
[边] 吃饭 [边] 做作业 indicates "eat dinner" (verb A) and "do homework" (verb B) are carried out concurrently
[在] [边吃饭，边做作业] indicates both actions are in a progressive tense (in the middle of doing it / in a continuous state of doing it)
You can think of "一边...一边" as a split screen. To me, it indicates the concurrence of two actions.
To your question specifically, it seems the second sentence is verbose and do not follow what colloquial description is from a native speaker point of view.
Alternatively, You can say something like,
to indicate the current continuous tense though.
我一边在吃饭，一边在做作业。that's wrong, it should be
我在边吃饭，边做作业, that's right. 在 = doing something, stressed doing something (2 things at the same time).
and 我一边吃饭，一边做作业。 that's right. 一边...一边 = stressed at the same time, do 2 things