"Look up" vs. "Look for" phrasal verb
Solution 1:
There is a difference between look up and look for.
The phrasal verb look up means to find in some kind of authoritative reference, like a dictionary or encyclopedia.
The phrasal verb look for means to try to find something, but not necessarily in an authoritative reference.
The problem is that I don't see any significant difference in meaning between search and try to find, so I would have to say that both pairings are correct.