If the system of equations 2x+y-z=5 2x-5y+ z= x+2y-5z=7 has infinitely many solutions, the…

Question

If the system of equations

$2x+y-z=5$

$2x-5y+\lambda z=\mu $

$x+2y-5z=7$

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

Accepted Answer

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

For infinite solution $\Delta=0={\Delta}_{1}={\Delta}_{2}={\Delta}_{3}$

$\Delta=\begin{vmatrix} 2 \quad  1 \quad  -1 \ 2 \quad  -5 \quad  \lambda \ 1 \quad  2 \quad  -5 \end{vmatrix}=0$

$\begin{gathered} \Rightarrow 2(25-2\lambda )-(-10-\lambda )-(4+5)=0 \ \ \Rightarrow 50-4\lambda +10+\lambda -9=0 \ \ \Rightarrow 51=3\lambda \Rightarrow \lambda =17 $

${\Delta}_{3}=\begin{vmatrix} 5 \quad 2 \quad 1 \ \mu \quad 2 \quad -5 \ 7 \quad 1 \quad 2 \end{vmatrix}=0$

$ \Rightarrow 2(-35-2\mu )-(14-\mu )+5(4+5)=0 \ \ \Rightarrow -70-4\mu -14+\mu +45=0 \ \ \Rightarrow -3\mu =39 \ \ \Rightarrow -\mu =13 \end{gathered}$

Now $(\lambda +\mu {)}^{2}+(\lambda -\mu {)}^{2}$

$\begin{gathered} (17+13{)}^{2}+(17-13{)}^{2} \ 900+16 \ =916 \end{gathered}$

Solutions · 1