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When I read math books, I see sentences such as

x=2 is a unique solution of the equation.

I interpret the word unique as only one. As far as I know, a is used for something arbitrary, while the is used for something unique. Then shouldn't we use the in front of unique? Or maybe I am misunderstanding the meaing of unique. Is the following sentence correct?

The Moon is a unique satellite of the Earth.

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The sentence

The Moon is a unique satellite of the Earth.

is grammatically correct, however, it is ambiguous and may not mean what you think it does.

If what you meant to say is that Earth only has a singular satellite then the way to say it is :

The Moon is the only satellite of the Earth.

If what you meant was that of all of Earth's satellites, the Moon is the only one of its kind, then

The Moon is a unique satellite of the Earth because we never see its far side.

Unique is a funny word and well, it's, unique (bad pun intended...)

One trick I use to figure out whether to use a or the is to look at the phrase in the absence of unique

1) The Moon is a satellite of the Earth.
2) The Moon is the satellite of the Earth.
3) The Moon is a unique satellite of the Earth.

#1 sounds good while #2 doesn't, therefore use a in #3

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    To elaborate on this answer: the English word unique has evolved to mean "very special", whereas the original French word means "one and only". – 200_success Dec 15 '15 at 5:07
  • Sorry, I have a problem understanding your sentence explaining unique satellite. What do you mean by the Moon is the only one of its kind? Does the sentence mean that there are many satellites of the Earth and the Moon is just one of them? Or does it mean that there is only one satellite in the universe? Both interpretations seem weird. Could you explain more? – Hwang Dec 15 '15 at 8:49
  • @Hwang, I've edited my answer in response to your question. It means that there are several satellites around Earth, and the Moon is different from the rest because... It does not mean the Moon is just one of them since if it was just one of them, it would not be unique. Unique in this sense means to be the only thing to have certain characteristics that make it standout. Hope that helps – Peter Dec 16 '15 at 6:11
  • If my comment helped you right now, then you would say: I understand. If you figured it out before I wrote my comment, the you would write I understood... Keep asking questions! – Peter Dec 16 '15 at 6:46
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A is being used as the Indefinite Article. This implies that there are numerous solutions available, and of those available solutions x=2 happens to be unique.

If it was the Definite Article the solution then the unique solution would also be the only solution.

The indefinite article in this sense illustrates an example.

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