Equations – Troubleshooting System of Equations with Aligned Coefficients

alignequationstext;

The part above the hrule is what I have when aligning the coefficients. What I would like to do now is to write \minand \text{s.a.} and also add the last line, i.e., x_i \geqslant 0, \forall i \in [7]. I am having some trouble doing so without messing up the alignment of the coefficients. Below I leave my code for the system ABOVE the hrule. Thanks for all the help in advance.

enter image description here

\documentclass[12pt]{report}
    \usepackage{amsmath,systeme}
    \begin{document}
    
    \begin{equation*}
        \sysdelim..\systeme{-10x_1+57x_2+9x_3+24x_4,
                            0.5x_1-5.5x_2-2.5x_3+9x_4+x_5=0, 
                            0.5x_1-1.5x_2-0.5x_3+x_4+x_6=0,
                            x_1+x_7 = 1}
    \end{equation*}
    
    \end{document}

Best Answer

Here's a solution that employs two array environments.

enter image description here

\documentclass[12pt]{report}
\usepackage{amsmath,amssymb,array}
\newcolumntype{C}{>{{}}c<{{}}} % for binary/relational operators
\begin{document}
\begin{equation*}
\setlength\arraycolsep{0pt}
\renewcommand\arraystretch{1.25}
\begin{array}{@{}c@{}} 
  \min \\[0.5ex] \text{s.a.} \\ \\ \\ \\
\end{array}
\quad
\begin{array}{  r *{7}{Cr} }
-10x_1 &+&  57x_2 &+&   9x_3 &+& 24x_4 \\[0.5ex]
0.5x_1 &-& 5.5x_2 &-& 2.5x_3 &+&  9x_4 &+& x_5 & &     & &     &=& 0 \\
0.5x_1 &-& 1.5x_2 &-& 0.5x_3 &+&   x_4 & &     &+& x_6 & &     &=& 0 \\
   x_1 & &        & &        & &       & &     & &     &+& x_7 &=& 1 \\
\multicolumn{15}{l}{\phantom{-10}x_i \geqslant 0 \ \forall i\in\{1,2,\dots,7\} }
\end{array}
\end{equation*}
\end{document}
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