documentclass[10pt]{amsart}
usetikzlibrary{calc,angles,positioning,intersections,quotes,decorations.markings}
usepackage{mathtools,array}

usepackage{pgfplots}
pgfplotsset{compat=1.11}


begin{document}

begin{gather*}
left{
begin{array}{@{}c@{}c@{}c@{}c@{}c}
-3x &   {}-{}   &   6y  &   {}={}   &   -15 \
3x  &   {}+{}   &   4y  &   {}={}   &   6
end{array}
right.
, \
left{
begin{array}{@{}r@{}c@{}l}
-3x - 6y    &   {}={}    &   -15 \
-2y         &   {}={}    &   -9
end{array}
right.
, \
left{
begin{array}{@{}r@{}c@{}l}
x + 2y  &   {}={}   &   5 \
y       &   {}={}   &   9/2
end{array}
right.
, \
left{
begin{array}{@{}r@{}c@{}l}
x   &   {}={}   &   -4 \
y   &   {}={}   &   9/2
end{array}
right.
.
end{gather*}


end{document}