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

Question

If the system of equations

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

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

Accepted Answer

$\begin{gathered} 11x+y+\lambda z=-5 \ 2x+3y+5z=3 \ 8x-19y-39z=\mu \ \Delta=0\Rightarrow |\begin{array}{ccc}11 \quad  1 \quad  \lambda \ 2 \quad  3 \quad  5 \ 8 \quad  -19 \quad  -39\end{array}|=0 \ 11(-39.3+19.5)-1(-39.2-40)+\lambda (-38-24)=0 \ =11(-117+95)-1(-118)-62\lambda =0 \ =-242+118=62\lambda \ \Rightarrow \lambda =-2 \ \Delta2=0 \ \Rightarrow |\begin{array}{ccc}11 \quad  1 \quad  -5 \ 2 \quad  3 \quad  3 \ 8 \quad  -19 \quad  \mu \end{array}|=0 \end{gathered}$

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

Solutions · 1