Þessi lausn er einstök.

Þessi means this.

In Þessi lausn er einstök, it points to a specific solution: this solution.

It is a demonstrative word, like this / that in English. In Icelandic, þessi changes form depending on gender, number, and case, but in this sentence þessi is the correct form for a feminine singular nominative noun.

Because þessi adds the meaning this.

• lausn = solution
• þessi lausn = this solution

If you remove þessi, the sentence becomes Lausn er einstök, which sounds incomplete in normal Icelandic unless used in a very special context. Icelandic usually needs some clearer marking here, such as:

• Lausnin er einstök = The solution is unique
• Þessi lausn er einstök = This solution is unique

Icelandic usually does not use a separate word like English the. Instead, definiteness is often added to the end of the noun.

For example:

• lausn = solution
• lausnin = the solution

But in this sentence, you already have þessi (this), so you do not need the. English also does this:

• this solution not normally
• this the solution

So þessi lausn is perfectly natural.

Lausn is a feminine noun.

That matters because Icelandic adjectives and demonstratives must agree with the noun they describe. So both þessi and einstök have forms that match a feminine singular nominative noun.

Here the pattern is:

• þessi → matches lausn
• einstök → also matches lausn

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

Because einstök has to agree with lausn, which is feminine singular nominative.

The adjective is based on einstakur, but it changes form depending on what noun it describes:

• einstakur = masculine singular nominative
• einstök = feminine singular nominative
• einstakt = neuter singular nominative

So:

• einstakur maður = a unique man
• einstök lausn = a unique solution
• einstakt tækifæri = a unique opportunity

In your sentence, lausn is feminine, so einstök is required.

Usually it means unique, but like many adjectives, its exact nuance depends on context.

It can suggest something like:

• unique
• exceptional
• special
• one of a kind

In Þessi lausn er einstök, the most direct translation is This solution is unique, but depending on context it might also imply that the solution is unusually good, special, or distinctive.

Er means is.

It is the present tense, 3rd person singular form of the verb vera (to be).

So:

• ég er = I am
• þú ert = you are
• hann / hún / það er = he / she / it is

In this sentence:

• Þessi lausn = subject
• er = is
• einstök = predicate adjective

Yes, in this sentence the word order is very similar to English:

• Þessi lausn = This solution
• er = is
• einstök = unique

So the structure is:

subject + verb + adjective

That said, Icelandic word order can be more flexible than English, and in main clauses it often follows a verb-second pattern. But this sentence is a very straightforward example and lines up neatly with English.

A rough English-friendly guide would be:

THESS-ih LURSN er AYN-stuk

A few important points:

• Þ / þ sounds like th in thing, not like th in this
• ss is just s
• au in lausn does not sound exactly like English ow or aw; Icelandic vowels are different, but an English approximation is acceptable at first
• ö in einstök is like a sound English does not quite have; something a bit like the vowel in some pronunciations of bird, but shorter and rounder

The best way to improve pronunciation is to hear native audio, because Icelandic spelling is fairly consistent, but some sounds are unfamiliar to English speakers.

Yes.

• Þessi lausn er einstök = This solution is unique
• Lausnin er einstök = The solution is unique

The difference is the same as in English:

• þessi lausn points to a solution more explicitly: this one
• lausnin simply means the solution

So if you are contrasting one solution with others, þessi lausn is especially natural.

It actually is a declined form already, but þessi appears in several forms that look the same on the surface.

For learners, that can be confusing. The important point is that demonstratives in Icelandic do decline for:

• gender
• number
• case

In this sentence, þessi is the form that fits the noun phrase feminine singular nominative.

So even though it may look simple, it is still part of Icelandic agreement grammar.