Triangles $ABC$ and $ADE$ have areas $1000$ and $3000$, respectively, with $B=(0,0)$, $C=(100,0)$, $D=(300,200)$, and $E=(310,210)$. What is the sum of all possible $x$-coordinates of $A$?

1. Let $A=(a,b)$. By the shoelace formula (15), $[ABC]=50b=\pm1000$, so $b=\pm20$.
2. $[ADE]=-5a+5b+500=\pm3000$, so $b-a=\dfrac{\pm3000-500}5$, giving $a=b-500$ or $a=b+700$.
3. The four values of $a$ are $-480,720,-520,680$, summing to $\boxed{400}$.