How to have a and/or system of equations?

Question

Something like this:

without too much stretching of the height of the system

I specifically want the word "or"/"and" to be inserted between the two lines (or more ?) of equation, using only cases is not a solution.

Answer 1

If I understood you question correctly, than the following MWE gives what you after:

\documentclass{article}
\usepackage{amsmath}
\begin{document}
[
  \left{\begin{aligned}
      x &= 1 \
      \text{or} \
      y &= 1 \
    \end{aligned}\right.
]
\end{document}

Answer 2

Plain TeX macro \cases doesn't add a space at the right when the second column is omitted.

$$
  \cases{ x = 1       \cr
          \hfil\rm or \cr
          y = 1
  }
$$
\bye

Answer 3

Update.

It turns out cases, suggested below, leaves a small space behind, at the right and a better solution is proposed based on aligned environment

A comparison with cases

\documentclass{article}
\usepackage{amsmath}
\usepackage{mleftright}
\begin{document}
\mleftright
\begin{equation}
  \left\{
    \begin{aligned}
      x &= 1 \\
      y &= 1 \\
    \end{aligned}
  \right.\tag{1}
\end{equation}
\end{document}

---

Usually, the cases environment is for this type of expressions

\documentclass{article}
\usepackage{amsmath}
\begin{document}
\[
  \begin{cases}
    x = 1 \\
    y = 1
  \end{cases}
\]
\end{document}

If you expect fractions and would like them to be typed in display style, use dcases instead

\documentclass{article}
\usepackage{mathtools}
\begin{document}
\[
 \begin{dcases}
   x = \frac{1}{2} \\
   y = 1
 \end{dcases}
\]
\end{document}