"Excuse me, are your children already here?"
Translation:Извините, ваши дети уже здесь?
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My answer was marked incorrect: Извините, твои дети уже здесь? However, there was nothing in the English sentence "Excuse me, are your children already here?" to indicate that formal speech needs to be used, or that the question is to multiple people, so I have reported, as I think твои should be accepted as an alternative to ваши.
твои has a singular 'you', ваши a plural 'you' (but both of these are inflected for the main noun being plural). 'твой' (and thus of course твои) is used when there's only 'one' of you, and when not particularly concerned with being very polite. If there's more than one of 'you', or there's reason to be polite, ваш (and ваши) are used.
So, ваш functions like "y'all's" but without the connotations of being a country bumpkin.
Since you ask about твой vs. ваши I am a bit unsure about whether you have understood the bit about these pronouns agreeing with the owned thing as well.