How to align two \begin{equation}\begin{cases}?

Question

The code below generates the following figure:

Is there a way to align these two equations, like this:

i.e., align at the beginning of the bracket (not the at the equal signs!)

I have consulted several other questions, but they either did not have "cases", or they did not have equations labels. Thus the code will not work with labelled equations.

Does anyone know a catch-all method for aligning these types of equations? Thanks in advance!

\documentclass{article}
\usepackage[utf8]{inputenc}
\usepackage{amsthm}
\usepackage{amsmath}
\usepackage{graphicx}

\begin{document}
    \begin{equation}
    \label{eqn:equation1}
    \begin{cases}
    a &= b^2 + c^2\\
    y(t) & = a
    \end{cases}
    \end{equation}
    now, the second equation
    \begin{equation}
    \label{eqn:equation2}
    \begin{cases}
    b &= \sin(b^2) + c^2\\
    y(t) & = b
    \end{cases}
    \end{equation}
\end{document}

Update: Please also assume that the equations are tagged

\begin{document}
    \begin{equation}
    \tag*{eq 1}
    \label{eqn:equation1}
    \begin{cases}
    a &= b^2 + c^2\\
    y(t) & = a
    \end{cases}
    \end{equation}
    now, the second equation
    \begin{equation}
    \tag*{eq 2}
    \label{eqn:equation2}
    \begin{cases}
    b &= \sin(b^2) + c^2\\
    y(t) & = b
    \end{cases}
    \end{equation}
\end{document}

Answer 1

You can use \intertext, from the amsmath package, inside an align environment:

This sort of example, I believe, is exactly why the \intertext command exits: it inserts a "normal" line of text between equations that have common alignment points.

Here's the full MWE:

\documentclass{article}
\usepackage[utf8]{inputenc}
\usepackage{amsthm}
\usepackage{amsmath}
\usepackage{graphicx}

\begin{document}
    \begin{align}
    \tag*{eq 1}
    \label{eqn:equation1}
    &\begin{cases}
    a &= b^2 + c^2\\
    y(t) & = a
    \end{cases}
    \intertext{now, the second equation}
    \tag*{eq 2}
    \label{eqn:equation2}
    &\begin{cases}
    b &= \sin(b^2) + c^2\\
    y(t) & = b
    \end{cases}
    \end{align}
\end{document}

Answer 2

You can set both using aligned, and add the largest right-hand side to the shorter construction (top) via a \phantom:

\documentclass{article}

\usepackage{mathtools}

\begin{document}

\begin{equation}
  \tag*{eq 1} \label{eqn:equation1}
  \left\{\begin{aligned}
    \mathrlap{a}\phantom{y(t)} &= b^2 + c^2 \\
                         y(t)  &= \mathrlap{a}
                                  \phantom{\sin(b^2) + c^2}
  \end{aligned}\right.
\end{equation}
now, the second equation
\begin{equation}
  \tag*{eq 2} \label{eqn:equation2}
  \left\{\begin{aligned}
    \mathrlap{b}\phantom{y(t)} &= \sin(b^2) + c^2 \\
                         y(t)  &= b
  \end{aligned}\right.
\end{equation}

\end{document}

Left-hand side adjustments follow a similar pattern of overlapping and \phantom placement to achieve a left-aligned look.