"I am in the east and my children are in the west."
Translation:Jestem na wschodzie a moje dzieci są na zachodzie.
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Because there is contrast. I am in one place, my children are in another place. If we all were in one place together, then you could use "i".
Rule of thumb: if in English you could change "and" to "whereas", "while" or "but", with the meaning being basically still the same, then in Polish you use "a".
"u" is almost only used for being 'at someone's place'.
"w" simply means "in", "inside", so it generally is used for being in closed spaces, buildings.
"na" is literally "on", but it's also used for being 'in/at' open spaces.
The "w"/"na" distinction is a generalization though, and there are exceptions, they aren't even that rare. For example you say "na poczcie" (at the post office), although the post office is a building.
na wschodzie, na zachodzie (kraju - of the country)
na Wschodzie, na Zachodzie (kuli ziemskiej - of the globe)
If you live in the New York State, you are in the east of the country
or in the eastern part of the country -
Jeżeli mieszkasz w Stanie Nowy Jork, jesteś na wschodzie kraju,
lub we wschodniej części kraju
If you live in California, you are in the west of the country
or in the western part of the country -
Jeżeli mieszkasz w Kaliforni, jesteś na zachodzie kraju,
albo w zachodniej części kraju
"My children are in the west" could be a sentence on its own, which helps see that "my children" is the subject, therefore it has to take Nominative - which excludes 'moich'.
The singular noun "dziecko" (a child) is neuter, therefore its plural is "not masculine-personal plural" (it's not from a masculine noun), for which the right form is "moje".