The value of k ∈ R, for which the following system of linear...

We have,

$3x - y + 4z = 3$

$x + 2y - 3z = - 2$

$6x + 5y + kz = - 3$

For infinitely many solutions,

\left|A\right| = 0$</p><p>$⇒ \left|$\begin{cases}3 & - 1 & 4 \\ 1 & 2 & - 3 \\ 6 & 5 & k\end{cases}$\right| = 0$</p><p>$⇒ 3 ( 2k + 15 ) + 1\left(k + 18\right) + 4\left(5 - 12\right) = 0$</p><p>$⇒ 3 ( 2k + 15 ) + k + 18 - 28 = 0$</p><p>$⇒ 7k + 35 = 0$</p><p>$⇒ k = - 5