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If the system of equations $x+4 y-z=\lambda, 7 x+9 y+\mu z=-3,5 x+y+2 z=-1$ has infinitely many solutions, then $(2 \mu+3 \lambda)$ is equal to :
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$\begin{aligned} & \Delta=\left|\begin{array}{ccc}1 & 4 & -1 \\ 7 & 9 & \mu \\ 5 & 1 & 2\end{array}\right|=0 \\ & \Rightarrow(18-\mu)-4(14-5 \mu)-(7-45)=0 \Rightarrow \mu=0 \\ & \Delta=\Delta_{\mathrm{x}}=\Delta_{\mathrm{y}}=\Delta_{\mathrm{z}}=0 \text { (For infinite solution) } \\ & \Delta_{\mathrm{x}}=\left|\begin{array}{ccc}\lambda & 4 & -1 \\ -3 & 9 & \mu \\ -1 & 1 & 2\end{array}\right|=0 \\ & \lambda(18-\mu)-4(-6+\mu)-1(-3+9)=0 \\ & 18 \lambda+24-6=0 \Rightarrow \lambda=-1\end{aligned}$
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