First slide

System of linear equations

Question

The values of k for which the system of equations $x + k y - 3 z = 0, 3 x + k y - 2 z = 0, 2 x + 3 y - 4 z = 0$ has a non-trivial solution is

Moderate

A

$\frac{21}{10}$
B

$\frac{31}{10}$
C

$- 5$
D

4

Solution

$T h e c o n d i t i o n f o r a h o m o g e n e o u s s y s t e m o f e q u a t i o n s t o h a v e a n o n - t r i v i a l s o l u t i o n i s Δ = 0$
$\Rightarrow$ $|\begin{matrix} 1 & k & - 3 \\ 3 & k & - 2 \\ 2 & 3 & - 4 \end{matrix}| = 0$
$\Rightarrow 1 (- 4 k + 6) - k (- 12 + 4) - 3 (9 - 2 k) = 0$
$\Rightarrow - 4 k + 6 + 8 k - 27 + 6 k = 0$
$\Rightarrow 10 k - 21 = 0$
$\Rightarrow k = \frac{21}{10}$

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