1957 AMC 12 Problem 41

Question

Given the system of equations

\begin{aligned} ax+(a-1)y&=1,\\ (a+1)x-ay&=1. \end{aligned}

For which one of the following values of

a

is there no solution for

x

and

y?

1

0

-1

\pm\dfrac{\sqrt2}{2}

\pm\sqrt2

A

2\times2

system can fail to have a unique solution when its coefficient determinant is zero

Compute

a(-a)-(a-1)(a+1)

The coefficient determinant is

\begin{aligned} &a(-a)-(a-1)(a+1)\\ &\quad=1-2a^2. \end{aligned}

It vanishes when

a=\pm\frac{\sqrt2}{2}.

For either value, the two coefficient rows are proportional but the two right sides are not in the same ratio, so the system is inconsistent.

Thus, the correct answer is D.

Problem 41 in other years