First slide

System of linear equations

Question

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

Moderate

A

$a + b = 2$
B

$a + b = a b$
C

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

$a + b = 0$

Solution

From (1) and (3)
$(a - 1) x - z = 0$
and from (2) and (3)
$(b - 1) y = z$
$∴ \frac{x}{\frac{1}{(1 - a)}} = \frac{y}{\frac{1}{(1 - b)}} = \frac{z}{1}$
Putting in (1) we get
$\frac{a}{1 - a} + \frac{1}{1 - b} + 1 = 0$
$\Rightarrow \frac{a}{1 - a} = - \frac{1}{1 - b}$
$\Rightarrow 1 - b = - 1 + a$
$\Rightarrow a + b = 2$

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