Aligning system of equations with zero coefficients

Like this answer.

\documentclass{article}
\usepackage{systeme}

\begin{document}
\sysdelim..\systeme{x_2      +2x_3          +2x_4         -2x_5    =1,
x_1                +3x_3                   +4x_5    =5, 
-x_1         +3x_2       +3x_3                   -10x_5    =4}
\end{document}

For the sake of variety, here's an implementation that uses Mico's answer from the post you originally linked only using array. Admittedly, it's not quite as elegant as the solution marmot provided. However, its output is exactly the same as what's produced by \systeme. Observe that both answers produce output that's more tightly spaced than what's shown in the screenshot you posted. The array-based solution is also somewhat more general than what's produced by \systeme, in the sense that it can handle fairly arbitrary variable names.

\documentclass{article}
\usepackage{array} % for "\newcolumntype" macro
\newcolumntype{C}{ >{{}}c<{{}} }

\usepackage{systeme}
\sysdelim..

\begin{document}
\[
\setlength\arraycolsep{0pt}
\renewcommand{\arraystretch}{1.21} % optional
\begin{array}{r *{5}{Cr}}
         & &  x_2 &+& 2x_3 &+& 2x_4 &-&  2x_5 &=& 1 \\
     x_1 & &      &+& 3x_3 & &      &+&  4x_5 &=& 5 \\
    -x_1 &+& 3x_2 &+& 3x_3 & &      &-& 10x_5 &=& 4
\end{array}
\]

%% for comparison, @marmot's answer which uses "\systeme"
\[
\systeme{      x_2 +2x_3 +2x_4 -2x_5 =1,
         x_1       +3x_3       +4x_5 =5, 
        -x_1 +3x_2 +3x_3      -10x_5 =4}
\]
\end{document}