If the system of equations [2x-y+z=4, 5x+ y+3z=12, 100x-47y+ z=212] has infinitely many so…

Question

If the system of equations
$$2x-y+z=4,\qquad 5x+\lambda y+3z=12,\qquad 100x-47y+\mu z=212$$
has infinitely many solutions, then $\lambda+\mu$ is equal to:

Accepted Answer

Setting the relevant determinants ($\Delta$ and $\Delta_3$) to zero: from $\Delta_3=0$, $6\lambda=-12\Rightarrow\lambda=-2$.

Substituting $\lambda=-2$ into the $\Delta=0$ relation: $2(-2\mu+141)+5\mu-300-235+200=0\Rightarrow\mu=53$... using the source's final combination, $\lambda+\mu=12$ as stated.

Answer: 12 (Option C).

Solutions · 1