First slide

System of linear equations

Question

Suppose a,b,c $\in R$ and $a b c, α \neq 0$ If the system of equations
$(a + α) x + α y + α z = 0 - - - - - (1)$
$α x + (b + α) y + α z = 0 - - - - - (2)$
$α x + α y + (α + c) z = 0 - - - - - - (3)$ has a non-trivial solution, then $α (\frac{1}{a} + \frac{1}{b} + \frac{1}{c}) i s =$

Moderate

A

-1
B

0
C

$a b c$
D

$a b + b c + c a$

Solution

From eqns 1 &2 we have
$a x - b y = 0 \Rightarrow a x = b y$
From eqns 2 &3 we have
$a x - c z = 0 \Rightarrow a x = c z$
$a x = b y = c z \Rightarrow x / 1 / a = y / 1 / b = z / 1 / c$
$S u b s t i t u t i n g x, y, z v a l u e s i n e q u a t i o n (1), w e g e t$
$\frac{a + α}{a} + \frac{α}{b} + \frac{α}{c} = 0 = 1 \Rightarrow α (\frac{1}{a} + \frac{1}{b} + \frac{1}{c}) = - 1$

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