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.

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