Skattskyldan og samsköttunin eru skýrar.

Because the subject is plural overall.

Even though skattskyldan and samsköttunin are each singular by themselves, they are joined by og (and), so together they form a plural subject:

• skattskyldan = the tax liability
• samsköttunin = the joint taxation
• skattskyldan og samsköttunin = the tax liability and the joint taxation

So Icelandic uses the plural form of vera (to be):

• er = is
• eru = are

Because it agrees with the two nouns joined by og.

In Icelandic, adjectives usually agree with the noun or nouns they describe in:

• gender
• number
• case

Here, the adjective is describing two things, so it must be plural. Since both nouns are feminine, the adjective is in the feminine plural nominative form:

• skýr = clear
• skýrar = clear, feminine plural

So:

• Skattskyldan og samsköttunin eru skýrar
  = The tax liability and the joint taxation are clear.

Because both nouns are feminine.

The two nouns are:

• skattskylda — feminine
• samsköttun — feminine

When a predicate adjective follows vera, it agrees with the subject. Since both subject nouns are feminine, the adjective becomes feminine plural:

• masculine plural: skýrir
• feminine plural: skýrar
• neuter plural: skýr

So skýrar is the correct agreement form here.

The dictionary forms are:

• skattskylda
• samsköttun

The forms in the sentence have the suffixed definite article attached:

• skattskylda → skattskyldan = the tax liability
• samsköttun → samsköttunin = the joint taxation

So the sentence uses the definite forms of both nouns.

Because Icelandic nouns belong to different declension patterns.

Both nouns are feminine, but feminine nouns do not all form the definite article in exactly the same way.

Here:

• skattskylda → skattskyldan
• samsköttun → samsköttunin

So the difference is not about meaning; it is about noun class and inflection pattern. This is very common in Icelandic: nouns of the same gender can still take different endings.

They are in the nominative case.

Why?

• They are the subject of the sentence.
• With vera (to be), the predicate adjective also normally appears in the nominative.

So:

• skattskyldan = nominative singular definite
• samsköttunin = nominative singular definite
• skýrar = nominative feminine plural

Yes. This is a very normal and straightforward structure:

• Skattskyldan og samsköttunin = subject
• eru = verb
• skýrar = predicate adjective

So the pattern is:

Subject + verb + complement

That is perfectly natural Icelandic. Icelandic often has verb-second word order in many contexts, but in a simple main clause like this, starting with the subject and then the verb is completely standard.

Yes, that kind of word order is possible in Icelandic, but it is more marked or stylistic.

The neutral, everyday order is:

• Skattskyldan og samsköttunin eru skýrar.

Putting skýrar first gives extra emphasis or a different style, and it would be less common as a basic textbook sentence.

So for normal use, the version you have is the best one to learn first.

Because skýr is not the correct agreement form for this subject.

The base form skýr is singular, and also not the right form for two feminine nouns together. Since the subject is:

• plural overall
• feminine

the adjective must be:

• skýrar

So:

• skýr = not correct here
• skýrar = correct

Usually you learn the gender with the noun as part of its vocabulary entry.

Some clues help:

• nouns ending in -a are often feminine, like skattskylda
• many nouns ending in -un are also feminine, like samsköttun

But these are tendencies, not absolute rules. The safest approach is:

• learn the noun
• learn its gender
• learn its definite form if possible

That helps you choose the right adjective and article endings later.

Yes. Skýr is a regular Icelandic adjective in the sense that it changes form to match the noun.

Here are some relevant forms:

• masculine singular: skýr
• feminine singular: skýr
• neuter singular: skýrt
• masculine plural: skýrir
• feminine plural: skýrar
• neuter plural: skýr

In this sentence, the needed form is feminine plural nominative, so we get skýrar.

This kind of agreement is one of the most important features of Icelandic grammar.

Grammatically, it treats them as two separate things, because the nouns are joined by og and the verb is plural:

• eru = are
• skýrar = clear, plural

So even if the two ideas are closely related in meaning, the grammar treats them as a plural pair. If Icelandic wanted to treat the whole thing as one singular concept, the sentence would have to be built differently.