First slide

System of linear equations

Question

Suppose $a, b \in R$ and $a, b \neq 1$ . If the system of equations $a x + y + z = 0, x + b y + z = 0, x + y + 2 z = 0$ has a non-trivial solution then

Moderate

A

$a + b = 2 a b$
B

$a + b = a b$
C

$a + \frac{1}{b} = 2$
D

$a + b = 0$

Solution

$T h e c o n d i t i o n f o r a h o m o g e n e o u s s y s t e m o f e q u a t i o n s t o h a v e a n o n - z e r o s o l u t i o n i s Δ = 0$
$\Rightarrow$ $|\begin{matrix} a & 1 & 1 \\ 1 & b & 1 \\ 1 & 1 & 2 \end{matrix}| = 0$
$\Rightarrow a (2 b - 1) - 1 (2 - 1) + 1 (1 - b) = 0$
$\Rightarrow 2 a b - a - 1 + 1 - b = 0 \Rightarrow a + b = 2 a b$

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