Alignment of multiple variables in a linear system

This is an intelligent solution with systeme package. Just type each term blindly, the package will arrange them automatically. For example, if you type 3z+y=10 in the first row, the term will be aligned as shown in the output image.

\documentclass[preview,border=12pt,varwidth]{standalone}
\usepackage{systeme,mathtools}
\begin{document}
\abovedisplayskip=0pt\relax% don't use this line in your production.
\[
\sysdelim.\}\systeme{
3z +y = 10,
x + y +  z = 6,
  3y - z = 13}
\quad
\!\begin{aligned}
x   &=  a\\
y   &=  b\\
z &=  c
\end{aligned}
\]
\end{document}

Here is one suggestion using environments from amsmath:

\documentclass{article}

\usepackage{amsmath}

\begin{document}

\begin{equation*}
  \left.
    \begin{alignedat}{4}
      x & {}+{} & y &       &    & = 1 \\
        &       & y & {}+{} & 2z & = 1 \\
      x &       &   & {}+{} & z  & = 0
    \end{alignedat}
  \right\} \qquad
    \begin{aligned}
     x &= 0\\
     y &= 1\\
     z &= 0
    \end{aligned}
\end{equation*}

\end{document}

You might consider adding \ or \quad before \right\} to get more space before the brace.

Just in case the amsmath isn't available to you, you can also use a couple of array environments for this type of thing

% arara: pdflatex

\documentclass{article}

\begin{document}

\[
  \left.
    \begin{array}{r@{}r@{}r@{}r@{}r@{}r@{}}
      x & {}+{} & y &       &    & {}= 1 \\
        &       & y & {}+{} & 2z & {}= 1 \\
      x &       &   & {}+{} & z  & {}= 0
    \end{array}
  \right\} \qquad
    \begin{array}{r@{}l}
     x &{}= 0\\
     y &{}= 1\\
     z &{}= 0
    \end{array}
\]

\end{document}