Al, Betty, and Clare split $\textdollar 1000$ among them to be invested in different ways. Each begins with a different amount. At the end of one year, they have a total of $\textdollar 1500$ dollars. Betty and Clare have both doubled their money, whereas Al has managed to lose $\textdollar100$ dollars. What was Al's original portion?
$\textbf{(A)}\ \textdollar 250 \qquad \textbf{(B)}\ \textdollar 350 \qquad \textbf{(C)}\ \textdollar 400 \qquad \textbf{(D)}\ \textdollar 450 \qquad \textbf{(E)}\ \textdollar 500$
For this problem, we will have to write a three-variable equation, but not necessarily solve it. Let $a, b,$ and $c$ represent the original portions of Al, Betty, and Clare, respectively. At the end of one year, they each have $a-100, 2b,$ and $2c$. From this, we can write two equations, marked by (1) and (2). 
\[a+b+c=1000\]
\[(1). \text{ }2a+2b+2c=2000\]
\[a-100+2b+2c=1500\]
\[(2). \text{ }a+2b+2c=1600\]
(Equations (1) and (2) are derived from each equation above them.)
Since all we need to find is $a,$ subtract the second equation from the first equation to get $a=400.$
Al's original portion was $\boxed{\textbf{(C)}\ \textdollar 400}$.