Do these words need an article? [closed]

Solution 1:

While x in "x axis" might be a noun, if it is then it's an attributive noun which functions as an adjective.

Consider a question and answer like

Which frame is largest?
The door frame is largest.

The main noun in your sentence is axis, and that does need an article, just like frame does in my example.

The same applies to your sentence about the variable, but we wouldn't generally phrase it as "The b variable": it would be "The variable b is the length". Here, b is a name, and the sentence is analogous to something like "The man Andrew is the president". That's a slightly defective analogy, but we certainly wouldn't say "The Andrew man is the president".

Condition in your third sentence is a countable noun. The subject could be in one of any number of conditions, resting or running or anything else. Because it's countable, it needs an article, and the indefinite article is appropriate.

Your reasoning for the last sentence is correct.