Translation:Is this free?
Why is the 的 at the end necessary for some "x is adj" phrases but not others?
I think the easiest way for an English speaker to understand 是…的 is to realize that the noun at the end of sentence has been left out because it is understood in the context.
When you want to equate two nouns, you say:
Noun1 是 noun2.
When you want to add an adjective description to a noun, you use:
adjective 的 noun.
So if you want to declare that a certain thing (noun) has a certain property (description), that becomes:
Noun1 是 adjective 的 noun2.
Example: This is a free book. 这是免费的书。
However, in reality, noun2 is often understood from the context and can therefore be omitted. This is the case in English as well.
Example: This is free. 这是免费的。
Note that when noun2 is understood, the English adjective looses the article that it needs before a noun ("a free book" became "free.") IN CONTRAST, the Chinese adjective must keep its 的 so that it can remain a descriptor of the understood noun (免费的书 becomes 免费的). Without the 的, 免费 would have to be interpretted as a noun, not a adjective, and the (ungrammatical) sentence would mean something nonsensical like "This book literally is the abstract state of having no cost."
And that's why you need the 的.
So when you see "Noun 是adjective 的“ remember that there's an understood noun2 at the end.
Because here 是 is used. My explanation is that 是 cannot be used with an adjective and 的 nominalises the adjective.
I agree. As far as I know, if you want to say "x is adj.", you can use any of these constructions:
noun + adjective
noun + 很 + adjective
noun + 是 + adjective + 的
Exception: Actually nobody says 这很免费. I think it is because nobody says something is very/more free of charge. One cannot think 很 can mean nothing but just functions as a pure connector here.
But these days netizens like to say “很 + [anything that is not an adjective]” for surprise or sarcasm — to mean something coheres with [anything] so unbelievably reasonably. ;-)
Does this mean: Is this free of charge? Or are there other interpretations possible? free = not occupied?