If the system of equations [aligned & 2 x-y+z=4 & 5 x+ y+3 z=12 & 100 x-47 y+ z=212 aligne…

Question

If the system of equations

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

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

Accepted Answer

$$\begin{aligned}
& \Delta=0 \Rightarrow\left|\[\begin{array}{ccc} 2 & -1 & 1 \ 5 & \lambda & 3 \ 100 & -47 & \mu \end{array}\end{aligned}$$\right|=0 \
\& 2(\lambda \mu+141)+(5 \mu-300)-235-100 \lambda=0 \ldots (1)\
\& \Delta\_3=0 \Rightarrow\left|$$\begin{array}{ccc} 2 & -1 & 4 \ 5 & \lambda & 12 \ 100 & -47 & 212 \end{array}$$\right|=0 \
\& 6 \lambda=-12 \Rightarrow \lambda=-2 \
\& \text { Put } \lambda=2 \text { in }(1) \
\& 2(-2 \mu+141)+5 \mu-300-235+200=0 \
\& \mu=53 \
\& \therefore 57
\]

Solutions · 1