Encantado, soy Luis. Igualmente, yo soy Carmen.

Breakdown of Encantado, soy Luis. Igualmente, yo soy Carmen.

yo
I
ser
to be
encantado
pleased to meet you
igualmente
likewise, you too (as a reply)

Questions & Answers about Encantado, soy Luis. Igualmente, yo soy Carmen.

If Carmen spoke first, would she also say Encantado?
No, she would say Encantada. Encantado literally means "delighted" or "charmed," so it acts like an adjective describing the speaker. Because Luis is a man, he uses the masculine form Encantado. A woman will always use Encantada.
Why does Luis just say soy Luis, but Carmen says yo soy Carmen?
In Spanish, you usually drop the subject pronoun (like yo, meaning "I") because the verb soy ("am") already makes it clear who is speaking. Luis does this normally. Carmen includes yo to add contrast or emphasis, which sounds like, "And I am Carmen."
How exactly does Igualmente work as a reply here?
Igualmente literally means "equally". In introductions, it is a very common and natural shorthand for "likewise" or "the pleasure is mine." It saves Carmen from having to repeat Encantada.

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