Analyzing the problem carefully, it is a dcases
environment of multi-line equations, whose second parts should be right justified (or right aligned, or \shoveright
, or \MoveEqRight
... you name it).
You need the nested multline
environment multilined[t]
for each equation, inside an \aligned
environment, inside a dcases
environment.
I am against this practice:
the brace should appear to the left with no indentation
Thus, I haven't included it in my code.
In the following example, Eq. (1) is not aligned while Eq. (2) is.

\documentclass{article}
\usepackage{mathtools}
\usepackage[nopar]{lipsum} % for dummy text only
\begin{document}
\lipsum[66]
%% FIRST EQ
\begin{equation}
\begin{dcases}
\begin{multlined}[t]
A_{PP}\sin\alpha_{P1} - A_{PS}\cos\alpha_{S1} +B_{PP} \sin\alpha_{P2} +B_{PS} \cos\alpha_{S2} =\\
=\sin\alpha_{P1},
\end{multlined}
\\
\begin{multlined}[t]
A_{PP}\cos\alpha_{P1} +A_{PS} \sin\alpha_{S1} -B_{PP}\cos\alpha_{P2} + B_{PS} \sin\alpha_{S2} = \\
=\cos\alpha_{P1},
\end{multlined}
\\
\begin{multlined}[t]
A_{PP}\gamma_{P1} \cos 2\alpha_{S1} +A_{PS} \gamma_{S1} \sin 2\alpha_{S1} +B_{PP} \gamma_{P2} \cos 2\alpha_{S2} -\\
- B_{PS} \gamma_{S2} \sin 2\alpha_{S2} = \gamma_{P1} \cos 2\alpha_{S1},
\end{multlined}
\\
\begin{multlined}[t]
-A_{PP}n_1 \gamma_{S1} \sin 2\alpha_{P1} +A_{PS} \gamma_{S1} \cos 2\alpha_{S1} +B_{PP} n_2 \gamma_{S2} \sin 2\alpha_{P2} + \\
+ B_{PS} \gamma_{S2} \cos 2\alpha_{S2} = n_1 \gamma_{S1} \sin 2\alpha_{P1}.
\end{multlined}
\end{dcases}
\end{equation}
\lipsum[66]
%% SECOND EQ
\begin{equation}
\begin{dcases}
\begin{aligned}
\begin{multlined}[t]
A_{PP}\sin\alpha_{P1} - A_{PS}\cos\alpha_{S1} +B_{PP} \sin\alpha_{P2} +B_{PS} \cos\alpha_{S2} =\\
=\sin\alpha_{P1},
\end{multlined}
\\
\begin{multlined}[t]
A_{PP}\cos\alpha_{P1} +A_{PS} \sin\alpha_{S1} -B_{PP}\cos\alpha_{P2} + B_{PS} \sin\alpha_{S2} = \\
=\cos\alpha_{P1},
\end{multlined}
\\
\begin{multlined}[t]
A_{PP}\gamma_{P1} \cos 2\alpha_{S1} +A_{PS} \gamma_{S1} \sin 2\alpha_{S1} +B_{PP} \gamma_{P2} \cos 2\alpha_{S2} -\\
- B_{PS} \gamma_{S2} \sin 2\alpha_{S2} = \gamma_{P1} \cos 2\alpha_{S1},
\end{multlined}
\\
\begin{multlined}[t]
-A_{PP}n_1 \gamma_{S1} \sin 2\alpha_{P1} +A_{PS} \gamma_{S1} \cos 2\alpha_{S1} +B_{PP} n_2 \gamma_{S2} \sin 2\alpha_{P2} + \\
+ B_{PS} \gamma_{S2} \cos 2\alpha_{S2} = n_1 \gamma_{S1} \sin 2\alpha_{P1}.
\end{multlined}
\end{aligned}
\end{dcases}
\end{equation}
\lipsum[66]
\end{document}
I came across the solution after two hours of playing around with align
, gather
, multline
, split
, alignat
, flalign
and all its variants; just before I was about to contact Lars Madsen (maintainer of the mathtools
package) for an upgrade. It had to be something with an automated format, in pure LaTeX style, not through the use of \quad
or \hspace
or \vphantom
or the like.
It is amazing how such a basic typesetting feature is so unknown. I mean, I couldn't find a solution on the whole Internet. It ought to be included in the the guides and tutorials. Multi-line equations are so common!
I knew there had to be more people demanding a solution to this, which I'm now sharing throughout the forum. It also proves how robust the coding of the mathtools
package is, responding successfully to this workaround.
\begin{array}{@{}c@{}}