First slide

System of linear equations

Question

If the system of equations , $x + y + z = 0$ , $a x + b y + z = 0$ , $b x + y + z = 0$ ,has a non-trivial solution, then

Moderate

A

$b^{2} = 2 b + 1$
B

$b^{2} = 2 b - 1$
C

$b - a = 0$
D

$b^{2} = 2 b$

Solution

From first and third equation
$(b - 1) x = 0$
If $b \neq 1, x = 0$
In this case $y + z = 0, b y + z = 0$
$\Rightarrow (b - 1) y = 0 \Rightarrow y = 0$
Thus , $z = 0$
That is, if $b \neq 1$ , then the system has only trivial solution.
$∴$ $b = 1$
From first and second equation
$(a - 1) x = 0 .$
If $a \neq 1, x = 0 a n d y + z = 0$
In this case the system has infinite number of solution
If $a = 0$ all the three equations become identical to $x + y + z = 0$ and the system has infinite number of solutions.
$b = 1 \Rightarrow (b - 1)^{2} = 0 \Rightarrow b^{2} = 2 b - 1$

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