Learn Polish: Prezydentowi zależy na gospodarce. - The president cares about the economy.

In Polish, expressing that you care about something uses the verb zależeć in an impersonal way. Think of it as meaning "it matters to...". The person who cares (the one it matters to) takes the Dative case. So, prezydent changes to the Dative form prezydentowi.

What case is gospodarce, and why?

It is in the Locative case. When you use zależeć to say what is important, it is always followed by the preposition na plus the Locative case. The dictionary form is gospodarka (economy), and the -ka ending softens to -ce in the Locative.

If multiple presidents cared about the economy, would the verb zależy change?

No, the verb stays exactly the same! Because the construction is impersonal (meaning "it matters"), the verb is locked in the third-person singular. You only change the person experiencing it. "The presidents care" would be Prezydentom zależy na gospodarce.

Does zależeć na work for "caring" in the sense of nursing someone back to health?

No, zależeć na strictly means "to care about" in the sense that something is important to you. If you want to talk about physically taking care of someone or nursing them, you should use the verb opiekować się.