If the system of equations gathered 2x+ y+3z=5 3x+2y-z=7 4x+5y+ z=9 gathered has infinitel…

Question

If the system of equations

$\begin{gathered} 2x+\lambda y+3z=5 \ 3x+2y-z=7 \ 4x+5y+\mu z=9 \end{gathered}$

has infinitely many solutions, then $({\lambda }^{2}+{\mu }^{2})$ is equal to :

Accepted Answer

$\begin{gathered} 2x+\lambda y+3z=5 \ 3x+2y-z=7 \ 4x+5y+\mu z=9 \ \Rightarrow \Delta=0={\Delta}_{1}={\Delta}_{2}={\Delta}_{3} $

$\begin{gathered} \Delta=|\begin{array}{ccc}2 \quad \lambda \quad 3 \ 3 \quad 2 \quad -1 \ 4 \quad 5 \quad \mu \end{array}|=0 \ \Rightarrow -4\lambda -3\lambda \mu +4\mu +31=0 \ {\Delta}_{1}=|\begin{array}{ccc}5 \quad \lambda \quad 3 \ 7 \quad 2 \quad -1 \ 9 \quad 5 \quad \mu \end{array}|=0\Rightarrow -9\lambda -7\lambda \mu +10\mu +76=0 \ {\Delta}_{2}=|\begin{array}{ccc}2 \quad 3 \quad 5 \ 3 \quad -1 \quad 7 \ 4 \quad \mu \quad 9\end{array}|=0\Rightarrow \mu +5=0\Rightarrow \mu =-5 \ {\Delta}_{3}=|\begin{array}{ccc}2 \quad \lambda \quad 5 \ 3 \quad 2 \quad 7 \ 4 \quad 5 \quad 9\end{array}|=0\Rightarrow \lambda +1=0\Rightarrow \lambda =-1 \end{gathered}$

$ 	herefore \mu =-5\lambda =-1 \ {\mu }^{2}+{\lambda }^{2} =25+1 \ =26 \end{gathered}$

Solutions · 1