If the system of equations gathered 2x-y+z=4 5x+ y+3z=12 100x-47y+ z=212 gathered has infi…

Question

If the system of equations

$\begin{gathered} 2x-y+z=4 \ 5x+\lambda y+3z=12 \ 100x-47y+\mu z=212 \end{gathered}$

has infinitely many solutions, then $\mu -2\lambda $ is equal to

Accepted Answer

$\begin{gathered} \Delta=0\Rightarrow |\begin{array}{ccc}2 \quad  -1 \quad  1 \ 5 \quad  \lambda  \quad  3 \ 100 \quad  -47 \quad  \mu \end{array}|=0 \ 2(\lambda \mu +141)+(5\mu -300)-235-100\lambda =0\ldots (1) \ {\Delta}_{3}=0\Rightarrow |\begin{array}{ccc}2 \quad  -1 \quad  4 \ 5 \quad  \lambda  \quad  12 \ 100 \quad  -47 \quad  212\end{array}|=0 \ 6\lambda =-12\Rightarrow \lambda =-2 \ \lambda =2(1) \ 2(-2\mu +141)+5\mu -300-235+200=0 \ \mu =53 \ 	herefore 57 \end{gathered}$

Solutions · 1