How would you manage to put a vertical bar right to a system of equations?

It is possible to solve your question using systeme package adding a matrix to the right (for example).

 %% Compile and read me!
\documentclass[a4paper,12pt]{article}
\usepackage{mathtools}
\usepackage{systeme}
\begin{document}
$\sysdelim{\{}{|}\systeme{t+x+y+z=4,x+y=2,z=2} \quad \begin{matrix} \text{my text} \\[.3em] \text{another text} \\[.3em] \text{my favorite}\end{matrix}$
\end{document}

ADDENDUM: This addendum is of the great user @Bernard (see your comment into your answer). Without him, I wouldn't have had the left-aligned text.

\documentclass[a4paper,12pt]{article}
\usepackage{mathtools}
\usepackage{systeme}
\begin{document}
$\sysdelim{\{}{|}\systeme{t+x+y+z=4,x+y=2,z=2} \quad \begin{matrix*}[l] & \text{my text} \\[.3em] &\text{another text} \\[.3em] &\text{my favorite}\end{matrix*}$
\end{document}

Two variants, also with systeme:

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

\begin{document}

  \[ \systeme{t + x + y + z = 4, x + y = 2, z = 2}\sysdelim| . \systeme{\mbox{} @ \text{Some text}, \mbox{}@ \text{Some more text},\mbox{} @\text{A last text}} \]
    \bigskip
  \[ \sysdelim\{| \systeme{t + x + y + z = 4, x + y = 2, z = 2\enspace}
  \enspace \begin{aligned} & \text{Some text} \\ & \text{Some more text} \\ & \text{A last text} \end{aligned}\]


\end{document} 

An improvement to Sebastiano's and Bernard's nice answers; it uses a single \systeme and the text is given at the end of each line.

This way, even vertically large texts are coped with, see the second (admittedly contrived) example.

The text is typeset in a zero width box, but the widths are measured and the maximum is computed. At the end I issue a suitable \hspace command to take care of the texts when centering the whole block.

\documentclass[a4paper,12pt]{article}
\usepackage{mathtools}
\usepackage{systeme}

\usepackage{showframe} % to show the page frame, remove it

\newlength{\stextwidth}
\newcommand{\stext}[1]{%
  \settowidth{\dimen0}{\quad\,#1}%
  \ifdim\dimen0>\stextwidth \global\stextwidth=\dimen0\fi
  \quad\makebox[0pt][l]{\quad\,#1}%
}


\begin{document}

\begin{equation*}
\global\stextwidth=0pt
\sysdelim{\{}{|}
\systeme{
  t+x+y+z=4\stext{my text},
  x+y=2\stext{another text},
  z=2\stext{my favorite}
}\hspace{\stextwidth}
\end{equation*}

\end{document}

The second (contrived) example

\documentclass[a4paper,12pt]{article}
\usepackage{mathtools}
\usepackage{systeme}

\usepackage{showframe} % for judging the centering

\newlength{\stextwidth}
\newcommand{\stext}[1]{%
  \settowidth{\dimen0}{\quad\,#1}%
  \ifdim\dimen0>\stextwidth \global\stextwidth=\dimen0\fi
  \quad\makebox[0pt][l]{\quad\,#1}%
}


\begin{document}

\begin{equation*}
\global\stextwidth=0pt
\sysdelim{\{}{|}
\systeme{
  t+x+y+z=4\stext{\parbox[t]{8em}{my text that is too long for one line\vspace{2ex}}},
  x+y=2\stext{another text},
  z=2\stext{my favorite}
}\hspace{\stextwidth}
\end{equation*}

\end{document}

If more than one of these systems are needed, it's better to define a macro:

\documentclass[a4paper,12pt]{article}
\usepackage{mathtools}
\usepackage{systeme}

\usepackage{showframe} % for judging the centering

\newlength{\stextwidth}
\newcommand{\stext}[1]{%
  \settowidth{\dimen0}{\quad\,#1}%
  \ifdim\dimen0>\stextwidth \global\stextwidth=\dimen0\fi
  \quad\makebox[0pt][l]{\quad\,#1}%
}
\newcommand{\tsysteme}[1]{%
  \sysdelim{\{}{|}%
  \global\stextwidth=0pt
  \systeme{#1}%
  \hspace{\stextwidth}%
}

\begin{document}

\begin{equation*}
\tsysteme{
  t+x+y+z=4\stext{my text},
  x+y=2\stext{another text},
  z=2\stext{my favorite}
}
\end{equation*}

\end{document}