\documentclass{jarticle}

\usepackage{amsmath,amssymb}

\usepackage{color}
\usepackage{jumoline}
\usepackage{proofread}
% \noproofreadmark   % uncomment when you typeset the final manuscript

% just as you like ...
\MidlineHeight=0.4ex
\UMOlineThickness=0.2ex

\begin{document}
\delspan{This is an old passage,} \addspan{And this is a revised one.}

\delspan{You will need to enclose some LaTeX commands with braces,
just like {$y = ax + b$}. }
\addspan{This workaroud is needed for both delspan and addspan.}

\delspan{%
 You will need a comment sign (%) just after the opening brace ({)
 if a line break comes at the beginning of the delspan'ed passage.
}

Unfortunately, delspan cannot be applied to equations whereas
addspan can be. As a workaroud, you may use delspanx and strikemath. 

\delspanx{%
\begin{align}
 \strikemath{\dot{x}} & \strikemath{= ax + bu}\\
 \strikemath{y} & \strikemath{= cx}
\end{align}
}%
\addspan{
\begin{align}
 x_{k+1} &= ax_k + bu_k\\
 y_k & = cx_k
\end{align}
}

Here, delspanx just removes its argument when noproofreadmark is
specified. Otherwise it does nothing. 

\end{document}