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\documentclass{minimal}
\usepackage{fancybox}
\newlength{\boxedeqnlen}
\newenvironment{equationbox}{\vspace{1em}\\%
\begin{Sbox}\setlength{\boxedeqnlen}{\linewidth}%
\addtolength{\boxedeqnlen}{-2\fboxsep}%
\addtolength{\boxedeqnlen}{-2\fboxrule}%
\begin{minipage}{\boxedeqnlen}%
\begin{equation}}%
{\end{equation}%
\end{minipage}%
\end{Sbox}\fbox{\TheSbox}\vspace{1em}\\}
\begin{document}
If we find the value of $x$ which solves the equation
\begin{equation}
2x-1=0
\end{equation}
we obtain
\begin{equationbox}
x=\frac{1}{2}
\end{equationbox}
where we employ the use of a \textit{fraction}.
\end{document}

Side note: I couldn't get /bgroup or /egroup to work, but I can't recall the exact problem. They seem to only be substitutes for { and } in certain scenarios.
Side note 2: The main problem with lrbox is that, while it does save text, it also formats it according to the currently activated environment, so if you do something like \begin{equation}\usebox{\thebox}\end{equation}, then the text in \thebox will still be formatted in the normal text mode.