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 $y?$

$1$

$0$

$-1$

$\pm\dfrac{\sqrt2}{2}$

$\pm\sqrt2$

Answer: D

Concepts:system of equations determinant

Difficulty rating: 1590

Small Hint:

A $2\times2$ system can fail to have a unique solution when its coefficient determinant is zero

Big Hint:

Compute $a(-a)-(a-1)(a+1)$

Solution:

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.

1957 AMC 12 Problem 41

Answer: D

Difficulty rating: 1590

Small Hint:

Big Hint:

Solution:

Problem 41 in Other Years