Let the system of equations x + 2 y + 3 z = 5 , 2 x + 3 y + z = 9 , 4 x + 3 y + λ z = μ have infinite number…

Given: $x+2y+3z=5$, $2x+3y+z=9$ and $4x+3y+\lambda z=\mu$.

For infinite solutions, $\Delta ={\Delta }_1={\Delta }_2={\Delta }_3=0$.

$\Rightarrow \Delta =\left|\begin{array}{ccc}1& 2& 3\\ 2& 3& 1\\ 4& 3& \lambda \end{array}\right|=0$

$\Rightarrow 3\lambda -3-4\lambda +8+18-36=0$

$\Rightarrow \lambda =-13$

Now, finding 

$\Rightarrow {\Delta }_1=\left|\begin{array}{ccc}5& 2& 3\\ 9& 3& 1\\ \mu & 3& -13\end{array}\right|=0$

$\Rightarrow \mu =15$

Now, finding

$\Rightarrow {\Delta }_2=\left|\begin{array}{ccc}1& 5& 3\\ 2& 9& 1\\ 4& \mu & \lambda \end{array}\right|$

$\Rightarrow {\Delta }_2=\left|\begin{array}{ccc}1& 5& 3\\ 2& 9& 1\\ 4& 15& -13\end{array}\right|$

$\Rightarrow {\Delta }_2=0$

And 

$\Rightarrow {\Delta }_3=\left|\begin{array}{ccc}1& 2& 5\\ 2& 3& 9\\ 4& 3& \mu \end{array}\right|$

$\Rightarrow {\Delta }_3=\left|\begin{array}{ccc}1& 2& 5\\ 2& 3& 9\\ 4& 3& 15\end{array}\right|$

$\Rightarrow {\Delta }_3=0$

For $\lambda =-13, \mu =15$ system of equation has infinite solutions.

$\Rightarrow \lambda +2\mu =17$.