\documentclass[12pt,a4paper]{article}
\usepackage{amsmath}
\usepackage{cases}
\begin{document}
Half-hearted recommendation
\begin{subnumcases} {\label{system}}
x-y = 1\label{x}\\
2\,x-y=3 \label{y}
\end{subnumcases}
Provide the solution to equation (\ref{system}) subtracting
(\ref{x}) from (\ref{y}).
But if you must have three separate numbers, you could put them in by
hand where cases expects a qualification
\begin{equation}
\label{system2}
\begin{cases}
x-y = 1 &\qquad \text{(a)}\\
2\,x-y=3&\qquad \text{(b)}
\end{cases}
\end{equation}
But those braces don't seem appropriate at all.
\begin{equation}
\label{system3}
\begin{aligned}
x-y &= 1 &\qquad& \text{(a)}\\
2\,x-y&= 3 &\qquad& \text{(b)}
\end{aligned}
\end{equation}
but probably best is this
\begin{subequations}
\label{system4}
\begin{align}
x-y &= 1 \label{xx}\\
2\,x-y&= 3 \label{yy}
\end{align}
\end{subequations}
combined eq.~(\ref{system4}).
What it sounds like you are asking for is this atrocity
\begin{align}
x-y &= 1 \tag{a}\\
& \\
2\,x-y&= 3 \tag{b}
\end{align}
which looks awful.
\end{document}