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I read this sentence on an educational website:

Now times have changed and you are ready for situations involving forces in two dimensions.

Shouldn't there be a comma before and, since the subject of the sentence has changed (things vs. you)?

Now times have changed, and you are ready for situations involving forces in two dimensions.

What is the exact comma rule for joining two clauses with and?

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According to some authorities, such as Purdue OWL, yes, a comma should be used before "and" in that sentence:

Use commas to separate independent clauses when they are joined by any of these seven coordinating conjunctions: and, but, for, or, nor, so, yet.

It's a compound sentence, as you identified, and should therefore have a comma.

That being said, a more reasonable guideline, as given at Grammartips.homestead.com, governing comma use is that they, like all punctuation, should be used to reduce or eliminate ambiguity. You can often eliminate the comma

if both independent clauses are quite short, especially if the two clauses are very closely related, and even more so if the subject of both clauses is the same, or

if only the first clause is quite short, especially if the two clauses are very closely related, and even more so if the subject of both clauses is the same.

Here is an example involving two short clauses conjoined with 'but':

John went to the store but he didn't buy anything.

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    Hmmm. I like this answer better than mine. Deleting, and +1. – T.E.D. Sep 21 '11 at 21:30
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    All very nicely put. I think the general trend is to use less punctuation these days, and it seems to me you recognise (and delicately endorse) that trend with your "...should be used to reduce...ambiguity. You can often eliminate the comma". – FumbleFingers Sep 21 '11 at 22:12
  • T.E.D.'s (deleted) answer adds: << I don't think there really is an "exact rule" [binding rule] that covers this situation and tells you whether a comma is required to be (or not to be) put there. It basically comes down to how the author wants the sentence to read. If they want a slight pause there, they should put in a comma. If not, then it should be left out. >> To which I'd further add that a comma before 'but' has the pragmatic effect of increasing contrast. And a comma before a coordinator where the main clauses are lengthy helps readability. – Edwin Ashworth Feb 7 '18 at 9:48

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