#align(left, $
& 1                   &&=& a_1  &+& a_2    &+& a_3    &+& a_4    \
& 1 + 2^3             &&=& 2a_1 &+& 2^2a_2 &+& 2^3a_3 &+& 2^4a_4 \
& 1 + 2^3 + 3^3       &&=& 3a_1 &+& 3^2a_2 &+& 3^3a_3 &+& 3^4a_4 \
& 1 + 2^3 + 3^3 + 4^4 &&=& 4a_1 &+& 4^2a_2 &+& 4^3a_3 &+& 4^4a_4
$)            


$ 
cases(
1                   & = thin a_1  &+ a_2    &+ a_3    &+ a_4   ,
1 + 2^3             & = thin 2a_1 &+ 2^2a_2 &+ 2^3a_3 &+ 2^4a_4,
1 + 2^3 + 3^3       & = thin 3a_1 &+ 3^2a_2 &+ 3^3a_3 &+ 3^4a_4,
1 + 2^3 + 3^3 + 4^4 & = thin 4a_1 &+ 4^2a_2 &+ 4^3a_3 &+ 4^4a_4,
)
$


$ cases(
     2y_1 +&&  y_2 =&&    4x_1\
      y_1 +&& 2y_2 =&&  -12x_2\
    -2y_1 +&& 3y_2 =&&     x_3
) $

$
     2y_1 +&&  y_2 =&&    4x_1\
      y_1 +&& 2y_2 =&&  -12x_2\
    -2y_1 +&& 3y_2 =&&     x_3
$