documentclass{amsart}
usepackage{amsmath}
usepackage{amsfonts}
usepackage{amssymb}
usepackage{amsthm}
usepackage{mathtools}


begin{document}

begin{equation*}
setlength{arraycolsep}{0pt}
left{
begin{array}{@{}rcl}
a^{6} + 2a^{3}b^{3} + b^{6} &{}={}& q^{2} \
4a^{3}b^{3} &{}={}& -dfrac{4}{27} , p^{3}
end{array}
right.
,
end{equation*}
begin{equation*}
setlength{arraycolsep}{0pt}
left{
begin{array}{@{}rcl}
a^{3} + b^{3} &{}={}& -q \
a^{6} - 2a^{3}b^{3} + b^{6} &{}={}& q^{2} + dfrac{4}{27} , p^{3}
end{array}
right.
,
end{equation*}
begin{equation*}
setlength{arraycolsep}{0pt}
left{
begin{array}{@{}rcl}
a^{3} + b^{3} &{}={}& -q \
(a^{3} - b^{3}){vphantom{a^{3} + b^{3}}}^{2}  &{}={}& q^{2} + dfrac{4}{27} , p^{3}
end{array}
right.
,
end{equation*}
begin{equation*}
setlength{arraycolsep}{0pt}
left{
begin{array}{@{}rcl}
a^{3} + b^{3}   &{}={}& -q \
a^{3} - b^{3}   &{}={}& sqrt{q^{2} + dfrac{4}{27} , p^{3}}
end{array}
right.
.
end{equation*}

end{document}