Equation tags with subequations

Question

Consider the following MWE:

\documentclass{article}

\usepackage{mathtools}

\begin{document}

\begin{align}
 \label{system}
  &\left\{
  \begin{aligned}
    30^{2} &= h_{C}^{2} + y^{2};\\
    40^{2} &= h_{C}^{2} + (50 - y)^{2};
  \end{aligned}
  \right.\\
  \ArrowBetweenLines[\Downarrow]
  &\left\{
  \begin{aligned}
    h_{C}^{2} &= 30^{2} - y^{2};\\
    h_{C}^{2} &= 40^{2} - (50 - y)^{2}.
  \end{aligned}
  \right.
\end{align}

\end{document}

I have two problems that I don't know how to solve:

• The equation tags are obviously not placed properly.
• I would like to label all four equation using the subequation environment in order to get "(1a)", "(1b)", "(1c)", and "(1d)".

Thank you in advance.

Answer 1

May be as a first try with my so-called null hack (of course not beautiful ;)), you can achieve what you want within a subequation environment by overloading the empheq package (of course along with the amsmath package).

\documentclass[10pt,a4paper]{article}
\usepackage{mathtools}
\usepackage{amsmath}
\usepackage[overload]{empheq}
\begin{document}
\begin{subequations}\label{e1}
    \begin{align}[left ={\empheqlbrace}]
    a = 1 &\label{e1a}\\
    b = 1 &\label{e1b}\\
    c = 1 &\label{e1c}
    \end{align}
    % the poor man's NULL hack :D
    \null\\
    \begin{align*}
     &\ArrowBetweenLines[\Downarrow]
     \end{align*}
    \begin{align}[left ={\empheqlbrace}]
    d = 1 &\label{e1d}\\
    e = 1 &\label{e1e}\\
    f = 1 &\label{e1f}
    \end{align}
\end{subequations}

\end{document}

which can give you with

Answer 2

\documentclass{article}
\usepackage[overload]{empheq}
\usepackage{mathtools}

\begin{document}

\begin{subequations}
\begin{align}[left = \empheqlbrace\,]
30^{2} &= h_{C}^{2} + y^{2};\\
40^{2} &= h_{C}^{2} + (50 - y)^{2};
\end{align}
\null\\[-5pt]
\begin{align*}
\\[-104pt]
&\phantom{sssssssssssssssssssssss}\ArrowBetweenLines[\Downarrow]
\\[-104pt]
\end{align*}
\null\\[-35pt]
\begin{align}[left = \empheqlbrace\,]
h_{C}^{2} &= 30^{2} - y^{2};\\
h_{C}^{2} &= 40^{2} - (50 - y)^{2}.
\end{align}
\end{subequations}

\end{document}