The package systeme offers different alignment options for the left and the right hand sides. Default is r,l, that is, unknowns right aligned and constant terms left aligned. This can be changed using \sysalign.
\documentclass{article}
\usepackage{systeme}
\begin{document}
[
\sysalign{r,r}% default is r,l
\systeme{x+y=2,x-y=-2}
]
\end{document}

Issue the \sysalign declaration in the preamble if you want it to hold for every system.
original answer (with older update)
It's a choice of the package to have the last column aligned left and, as far as I can see, there's no provision for changing it by setting an option.
You can change it by redefining the command \SYS@makesyspreamble@i that's responsible for setting the alignment in columns; the simplest way is to use regexpatch:
\documentclass{article}
\usepackage{systeme}
\usepackage{regexpatch}
\makeatletter
\xpatchcmd{\SYS@makesyspreamble@i}
{$##$\hfil\null}% left alignment
{\hfil$##$\null}% right alignment
{}{}
\makeatother
\begin{document}
[
\systeme{x+y=2,x-y=-2}
]
\end{document}

Should you need to switch between left and right alignment, here's how to do it.
\documentclass{article}
\usepackage{systeme}
\usepackage{regexpatch}
\makeatletter
\xpatchcmd{\SYS@makesyspreamble@i}
{$##$\hfil\null}
{\span\SYSX@afterequalalignment\null}
{}{}
\newcommand{\SYSX@afterequalalignmentleft}{$##$\hfil}% default
\newcommand{\SYSX@afterequalalignmentright}{\hfil$##$}% default
\newcommand{\rightalignafterequal}{%
\let\SYSX@afterequalalignment\SYSX@afterequalalignmentright
}
\newcommand{\leftalignafterequal}{%
\let\SYSX@afterequalalignment\SYSX@afterequalalignmentleft
}
\leftalignafterequal % default
\makeatother
\begin{document}
[
\systeme{x+y=2,x-y=-2}
]
[
\rightalignafterequal
\systeme{x+y=2,x-y=-2}
]
\end{document}
The \rightalignafterequal declaration obeys the usual scoping rules, so you can issue it in the preamble if you want all systems to have right alignment.

UPDATE
The most recent version of systeme no longer uses @ for private macros, so the patch should be modified.
\documentclass{article}
\usepackage{systeme}
\usepackage{regexpatch}
\chardef\savedunderscorecatcode=\catcode\_ \catcode_=11
\xpatchcmd{\SYS_makesyspreamble_i}
{$##$\hfil\null}
{\span\SYSX_afterequalalignment\null}
{}{}
\newcommand{\SYSX_afterequalalignmentleft}{$##$\hfil}% default
\newcommand{\SYSX_afterequalalignmentright}{\hfil$##$}% default
\newcommand{\rightalignafterequal}{%
\let\SYSX_afterequalalignment\SYSX_afterequalalignmentright
}
\newcommand{\leftalignafterequal}{%
\let\SYSX_afterequalalignment\SYSX_afterequalalignmentleft
}
\leftalignafterequal % default
\catcode`_=\savedunderscorecatcode
\begin{document}
[
\systeme{x+y=2,x-y=-2}
]
[
\rightalignafterequal
\systeme{x+y=2,x-y=-2}
]
\end{document}