When it's clear from the context that the adjective modifies both nouns, you use the singular form: "русский язык и культура", or "русская культура и язык". In this case, the gender of the adjective corresponds to the first noun.
When the context is ambiguous and it's not clear whether the adjective modifies both nouns or only one of them, you use the plural form. For instance, if you say "красивая женщина и мужчина", it may be understood as "an attractive woman and (just) a man". If you want to avoid ambiguity and say that both of them are attractive you use the plural adjective: "красивые женщина и мужчина".