First slide

System of linear equations

Question

An ordered pair (a,b) for which the system of linear equations $(1 + a) x + b y + z = 2, a x + (1 + b) y + z = 3, a x + b y + 2 z = 2$ has a unique solution is

Moderate

A

$(1, - 3)$
B

$(- 3, 1)$
C

$(2, 4)$
D

$(- 4, 2)$

Solution

$F o r u n i q u e s o l u t i o n o f a n o n - h o m o g e n e o u s s y s t e m o f e q u a t i o n s w e m u s t h a v e Δ \neq 0 . \Rightarrow |\begin{matrix} 1 + a & b & 1 \\ a & 1 + b & 1 \\ a & b & 2 \end{matrix}| \neq 0 a p p l y R_{1} \Rightarrow R_{1} - R_{2}, R_{2} \Rightarrow R_{2} - R_{3} \Rightarrow |\begin{matrix} 1 & - 1 & 0 \\ 0 & 1 & - 1 \\ a & b & 2 \end{matrix}| \neq 0 \Rightarrow 1 (2 + b) + 1 (0 + a) + 0 \neq 0 \Rightarrow a + b + 2 \neq 0 \Rightarrow O n l y (2, 4) s a t i s f i e s t h e c o n d i t i o n .$

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