Equation numbering in systeme package

For automatic equation numbering, with the equation numbers -- surrounded by parentheses -- placed to the right of the equations, it suffices to insert @(*) immediately after the first equation. If you require more separation -- say, \qquad -- you could insert @\qquad(*) instead.

\documentclass{article}
\usepackage{amsmath,systeme}

\begin{document}
\begin{gather*}
\systeme{3x+y+2z=6,x+3y+2z=-6,x+y+z=1} \\[2ex]        % no numbering
\systeme{3x+y+2z=6@(*), x+3y+2z=-6, x+y+z=1} \\[2ex]  % default separation
\systeme{3x+y+2z=6@\qquad(*), x+3y+2z=-6, x+y+z=1}    % separation by '\qquad'
\end{gather*}
\end{document}

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Addendum, to address the OP's follow-up question: To have the equation numbers produced by \systeme pick up the value of the equation counter, I suggest you insert the instructions

\renewcommand{\SYSeqnum}{\value{equation}}
\sysautonum{(**)}

in the preamble (after loading the systeme package) and omit the @(*) directives mentioned above. I.e., something like this:

\documentclass{article}
\usepackage{systeme}
\renewcommand{\SYSeqnum}{\value{equation}}
\sysautonum{(**)}

\begin{document}
\addtocounter{equation}{5}\theequation
\[
\systeme{3x+y+2z=6, x+3y+2z=-6 , x+y+z=1 }
\]

\stepcounter{equation}\theequation
\[
\systeme{3x+y+2z=6  , 
         x+3y+2z=-6 , 
         x+y+z=1    }
\]
\theequation
\end{document}

With this setup, the first three systeme equations will be numbered 6, 7, and 8, and the next three systeme equations will be number 10, 11, and 12.

---

Or, take the following example:

\documentclass{article}
\usepackage{amsmath,systeme}
\renewcommand{\SYSeqnum}{\value{equation}}
\sysautonum{(**)}
\begin{document}
\begin{gather} % create 7 dummy numbered equations:
a \\ b \\ c \\ d \\ e \\ f \\ g 
\end{gather}
\[
\systeme{3x+y+2z=6, x+3y+2z=-6 , x+y+z=1 }
\]
\end{document}

With this setup, the three systeme equations will be numbered 8, 9, and 10.

---

Second addendum, to address the topic of how to create cross-references to equations within a systeme system (pun intended). It's important to note that the standard \label-\ref mechanism doesn't work for such equations. The following MWE is based on the description of the systeme's approach, on p. 7 of the package's user guide.

\documentclass{article}
\usepackage{systeme}
\renewcommand{\SYSeqnum}{\value{equation}} 
\sysautonum{(**)}

\begin{document}
\setcounter{equation}{15}
\systeme{3x+y+2z=6 @\gdef\foo{**}(**),  % see p. 7 of the package's user guide
         x+3y+2z=-6@\gdef\bar{**}(**), 
         x+y+z=1   @\gdef\baz{**}(**)}      

Cross-references to equations (\foo) and (\baz).
\end{document}

This MWE outputs "Cross-references to equations (16) and (18)."

If it is possible to have numbers for the system equations not to be placed at the right margin, here is a simple solution:

\documentclass{article}
\usepackage{amsmath}
\usepackage{systeme}
\usepackage{linegoal} 

\begin{document}

\setcounter{equation}{10}

\[ \sysautonum*{\stepcounter{equation}\hbox{(\theequation)}}
\systeme{3x+y+2z=6,x+3y+2z=-6,x+y+z=1} \]%

\[ \sysautonum*{\stepcounter{equation}\hbox{(\theequation)}}
\systeme{3x+y+2z=6,x+3y+2z=-6,x+y+z=1} \]

\begin{equation}
  a + c = b +d
\end{equation}

\end{document}