First slide

System of linear equations

Question

The values of k for which the system of equations $\begin{matrix} x + k y - 3 z = 0, 3 x + k y - 2 z = 0, 2 x + 3 y - 4 z = 0 \end{matrix}$ has a non-trivial solution is (are)

Moderate

A

$\frac{21}{10}$
B

$\frac{31}{10}$
C

$- 5$
D

$4$

Solution

From (1) and (2), $2 x + z = 0 \Rightarrow 2 x = - z$
From (3), we get $3 y = 5 z$
$∴$ $x = - \frac{1}{2} z, y = \frac{5}{3} z$
Substituting this in (1) we get
$- \frac{1}{2} z + \frac{5 k}{3} z - 3 z = 0 . \Rightarrow (\frac{5 k}{3} - \frac{7}{2}) z = 0$
Now , $z \neq 0 \Rightarrow k = \frac{21}{10}$

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