Let λ be a real number for which the system of linear equations x +...

The system of equations ${a}_{1}x + {b}_{1}y + {c}_{1}z + {d}_{1} = 0, {a}_{2}x + {b}_{2}y + {c}_{2}z + {d}_{2} = 0$ and ${a}_{3}x + {b}_{3}y + {c}_{3}z + {d}_{3} = 0$ have a infinitely many solution, if

D = \left|\begin{cases}{a}_{1} & {b}_{1} & {c}_{1} \\ {a}_{2} & {b}_{2} & {c}_{2} \\ {a}_{3} & {b}_{3} & {c}_{3}\end{cases}\right| = 0, {D}_{1} = \left|\begin{cases}{d}_{1} & {b}_{1} & {c}_{1} \\ {d}_{2} & {b}_{2} & {c}_{2} \\ {d}_{3} & {b}_{3} & {c}_{3}\end{cases}\right| = 0, {D}_{2} = \left|\begin{cases}{a}_{1} & {d}_{1} & {c}_{1} \\ {a}_{2} & {d}_{2} & {c}_{2} \\ {a}_{3} & {d}_{3} & {c}_{3}\end{cases}\right| = 0 and {D}_{3} = \left|\begin{cases}{a}_{1} & {b}_{1} & {d}_{1} \\ {a}_{2} & {b}_{2} & {d}_{2} \\ {a}_{3} & {b}_{3} & {d}_{3}\end{cases}\right| = 0 .

Thus, for infinitely many solutions, we have $D = 0$

⇒ \left|\begin{cases}1 & 1 & 1 \\ 4 & \lambda & - \lambda \\ 3 & 2 & - 4\end{cases}\right| = 0

$⇒ 1\left(- 4\lambda + 2\lambda\right) - 1\left(- 16 + 3\lambda\right) + 1\left(8 - 3\lambda\right) = 0$

$⇒ - 2\lambda + 16 - 3\lambda + 8 - 3\lambda = 0$

$⇒ 24 - 3\lambda = 0$

$⇒ \lambda = 3$

And, $\lambda = 3$ satisfies the equation ${\lambda}^{2} - \lambda - 6 = 0 .$