A : Where were you at/where were you(what were you doing) when I called you on the phone?
B : I was just having dinner with my wife.
A : So, that was the reason you didn't pick up my phone call.
Like in this context, can I use "where were you at" or "where were you" to mean the same thing "what were you doing" ?
I guess "where were you at" can, but "where were you" can't.
" The intended meaning is the same either way. " by adding information about what they were doing. That doesn't mean that those expressions have the same meaning.
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"Where were you at?" is a non-standard way of saying "Where were you?" The intended meaning is the same either way.
Neither of those means "What were you doing?"
Nevertheless, in the given context a person is likely to answer "Where were you?" by adding information about what they were doing.
That doesn't mean that those expressions have the same meaning.
CJ
fire1can I use "where were you at
You can if you intend to speak in a non-standard way and sound uneducated. This is because at is superfluous.
fire1A : So, that was the reason you didn't pick up my phone call.
You should replace the yellow part with answer