If the system of equations \(11x+y+\lambda z=-5\\ 2x+3y+5z=3\\ 8x-19y-39z=\mu\) has infinitely many solutions, then \({\…

\(
\begin{aligned}
&11x+y+\lambda z=-5\\
&2x+3y+5z=3\\
&8x-19y-39z=\mu
\end{aligned}
\)

For infinite solutions.

\(
\begin{aligned}
&D=
\left|\begin{array}{ccc}
11 & 1 & \lambda\\
2 & 3 & 5\\
8 & -19 & -39
\end{array}\right|
=0\\
&\Rightarrow 11(-117+95)-1(-78-40)+\lambda(-38-24)\\
&\Rightarrow 11(-22)+118-\lambda(62)=0\\
&\Rightarrow 62\lambda=118-242\\
&\Rightarrow \lambda=\frac{-124}{62}=-2\\[4pt]
&D_1=
\left|\begin{array}{ccc}
-5 & 1 & -2\\
3 & 3 & 5\\
\mu & -19 & -39
\end{array}\right|
=0\\
&\Rightarrow -5(-117+95)-1(-117-5\mu)-2(-57-3\mu)=0\\
&\Rightarrow -5(-22)+117+5\mu+114+6\mu=0\\
&\Rightarrow 11\mu=-110-231=-341\\
&\Rightarrow \mu=-31\\[4pt]
&\lambda^4-\mu=(-2)^4-(-31)=16+31=47
\end{aligned}
\)