1957 AHSME Problem 41

Given the system of equations ax + (a - 1)y = 1 (a + 1)x - ay = 1 For which one of the following values of a is there no solution x and y?

\textbf{(A)}\ 1\qquad \textbf{(B)}\ 0\qquad \textbf{(C)}\ - 1\qquad \textbf{(D)}\ \frac {\pm \sqrt {2}}{2}\qquad \textbf{(E)}\ \pm\sqrt{2}

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.