Let α, β and γ be real numbers such that the system of linear...

x + 2y + 3z = \alpha$</p><p>$4x + 5y + 6z = \beta$</p><p>$7x + 8y + 9z = \gamma - 1$</p><p>The system of linear equations is consistent it means it has unique solution or infinite solutions</p><p>Here,$\Delta = \left|$\begin{cases}1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9\end{cases}$\right| = 0$</p><p>Hence, there are infinitely many solutions.</p><p>If equations have infinitely many solutions, then the equations are linearly connected, i.e.,${L}_{1} + \lambda{L}_{2} = {L}_{3}$</p><p>$⇒ \left(x + 2y + 3z - \alpha\right) + \lambda ( 4x + 5y + 6z - \beta ) = 7x + 8y + 9z - \gamma + 1$</p><p>$\dfrac{1 + 4\lambda}{7} = \dfrac{2 + 5\lambda}{8} = \dfrac{3 + 6\lambda}{9} = \dfrac{\alpha + \lambda\beta}{\gamma - 1}$</p><p>$\dfrac{1 + 4\lambda}{7} = \dfrac{2 + 5\lambda}{8} ⇒ \lambda = - 2$</p><p>Also,$\dfrac{1 + 4\lambda}{7} = \dfrac{\alpha + \lambdaB}{\gamma - 1}$</p><p>$⇒ - 1 = \dfrac{\alpha - 2\beta}{r - 1}$</p><p>$⇒ \alpha - 2\beta + \gamma = 1</p><p>Now, \left|M\right| = \left|\begin{cases}\alpha & 2 & \gamma \\ \beta & 1 & 0 \\ - 1 & 0 & 1\end{cases}\right|

$= \alpha + 2 ( - \beta ) + \gamma ( 1 ) = \alpha - 2\beta + \gamma = 1$