If the system of equations [[gathered 11 x+y+ z=-5 2 x+3 y+5 z=3 8 x-19 y-39 z= gathered]]…

Question

If the system of equations

$$\[\begin{gathered} 11 x+y+\lambda z=-5 \ 2 x+3 y+5 z=3 \ 8 x-19 y-39 z=\mu \end{gathered}$$\]

has infinitely many solutions, then $$\lambda^4-\mu$$ is equal to :

Accepted Answer

$$\begin{aligned}
& 11 x+y+\lambda z=-5 \
& 2 x+3 y+5 z=3 \
& 8 x-19 y-39 z=\mu \
& \Delta=0 \Rightarrow\left|\[\begin{array}{ccc} 11 & 1 & \lambda \ 2 & 3 & 5 \ 8 & -19 & -39 \end{array}\end{aligned}$$\right|=0 \
\& 11(-39.3+19.5)-1(-39.2-40)+\lambda(-38-24)=0 \
\& =11(-117+95)-1(-118)-62 \lambda=0 \
\& =-242+118=62 \lambda \
\& \Rightarrow \lambda=-2 \
\& \Delta 2=0 \
\& \Rightarrow\left|$$\begin{array}{ccc} 11 & 1 & -5 \ 2 & 3 & 3 \ 8 & -19 & \mu \end{array}$$\right|=0
\]

$$\begin{aligned}
& 11(3 \mu+57)-1(2 \mu-24)-5(-38-24)=0 \
& 33 \mu+627-2 \mu+24+310=0 \
& \mu=-31 \
& \Rightarrow \lambda^4-31 \
& =16+31 \
& =47
\end{aligned}$$

Solutions · 1