the equation (A) vs Equation (A)

Question

Quick question about grammar in English. Consider the equation $$x^2=c \tag{A}$$ Do we say that:

1. For every $c<0$, the equation (A) has no real solution.
2. For every $c<0$, Equation (A) has no real solution.

Detailed explanations other than "1) is correct" are welcome.

Answer 1

You should say "For every $c>0$, equation (A) has no real solution."

You should not use an article ("the" or "a") when you refer to something by name or label. This is a general principle: you can talk about "the president of the United States", but once you add a name it is "President Biden", not "the President Biden".

I personally prefer not to capitalise "equation (A)", but IMO this is a matter of taste; whether you use "equation (A)" or "Equation (A)", be consistent.

In general, you could also omit "equation" entirely, and just say e.g. "if $c$ is negative, (A) has no real solution." But I prefer not to have a mathematical expression and an equation number right next to one another, so I think it's better to include it in this case.