Learn Spanish (Latin America): Tú eres inteligente y tienes razón.

In English, we say someone is right, but in Spanish, being right is treated as something you possess. You literally say you have reason (tienes razón). This is a very common expression where Spanish uses tener (to have) for a mental state instead of the verb to be.

Why do we use eres for 'are smart' but tienes for 'are right'?

Being smart is considered a core characteristic of who you are, so Spanish uses eres, a form of the verb ser (to be). But being right is expressed with a tener (to have) phrase. So in this sentence, you are smart (eres inteligente) but you have rightness (tienes razón).

If I am speaking to a woman, do I need to change inteligente to inteligenta?

No, inteligente stays exactly the same. Adjectives that end in the letter e in Spanish do not change endings depending on whether you are describing a male or a female.

Why is the word tú included at the beginning, but not before tienes?