Can we use "with" instead of "by" after passive voice?
For instance, "it can be done by the aid of you". In the above example, can we apply "with" in lieu of "by"?
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This is a context where preposition usage has changed over time...
(I multiplied the aid values by 4 so one chart shows the same preference shift for both words at once.)
It's a risky business assigning precise and consistent meanings to English prepositions. In the specific context of linking a "result" (or method of achieving a result) to some relevant factor, by and with (plus through and possibly others) are "close enough". It's just that with is currently much more common for the exact context.
So you should note that the actual choice of preposition for OP isn't really a matter of semantics or grammar - it's just idiomatically established preference (which may slowly change). By further implication, although there will be some discernible patterns, the inconvenient truth is many of these idiomatically established usages simply have to be learned by rote (but not ...learned with rote! :). Just don't fall into the trap of assuming there's only one "correct" preposition for any given context (sometimes that's true, but often not).
Note that the specific fact of being "passive" doesn't affect OP's choice of preposition - exactly the same choice applies with all permutations of...
1: It was done [by / with / through] your [aid / help / assistance]
2: I did it [by / with / through] your [aid / help / assistance]
Your sentence is not grammatical. Things cannot be done by aid although they might be done by aiding somebody.
Your options are to write either:
It can be done with your aid
It can be done by you.
The first sentence is more likely than the second; nearly all English speakers would use the active voice: You can do it.
It can be done by the aid of you.
This sentence is weird and doesn't work well with passive voice unless you really mean "The aid of you" is the only thing that does "it".
You probably don't mean this--it's not just "the aid of you" that is doing anything. You more likely mean "the aid of you" plus some other things (the people/etc. needing aid), and all those things together can do "it".
So that's why you'd use with. By is a "replacement", with is an "addition".