Learn Swedish: Både hyran och räkningen är betalda. - Both the rent and the bill are paid.

In this sentence, the past participle betalda (paid) acts like an adjective describing the subject. Because the subject consists of two things (hyran and räkningen), it is plural. Just like regular adjectives, past participles get an -a added to the end when they are in the plural form.

Why can't I use the word betalat here, like in jag har betalat?

Betalat is the supine form, which is only used after the verb har (have) to say what someone has done (active). In this sentence, we are using är (are) to describe the condition of the rent and bill (they are in a paid state). For this, you need the past participle (betald/betalt/betalda), which changes its ending to match the subject just like an adjective.

How exactly do you say "both... and" in Swedish?

You use the pair både ... och. It works exactly like "both ... and" in English. You will almost always see både paired up with och later in the sentence.

Why are hyra and räkning in their definite forms (hyran, räkningen)?