Add equation name underneath equation number

Add equation name underneath equation number

I am wondering if it is possible (and if so how) to add names to equations
just underneath the equation number. I am just using the standard equation
environment:
\begin{equation}
\label{eqn:eqn_label}
y=mx+c
\end{equation}
to display the equation centered in the middle of the page, with the
equation number flushed to the right hand side. I need to put some text
immediately below the equation number to serve as the equation's name. The
text should be horizontally right-aligned with the equation number and
should not affect the latter's vertical alignment with the equation; i.e.
the equation and equation number should still be vertically in line with
each other.
I have tried using the \tag command, but realized that this not only
replaces the equation number, which I do not want, but also causes the
equation number counter to skip that particular equation. Ideally, the
equation should still function as a regular numbered equation, that can be
number-referenced (via \eqref or other), but with the addition of some
text below the equation number.
Any help will be much appreciated.