# If the system of linear equations

Question:

If the system of linear equations

$x+y+z=5$

$x+2 y+2 z=6$

$x+3 y+\lambda z=\mu,(\lambda, \mu \in R)$, has infinitely many

solutions, then the value of $\lambda+\mu$ is :

1. 12

2. 10

3. 9

4. 7

Correct Option: , 2

Solution:

$x+3 y+\lambda z-\mu=p(x+y+z-5)+$

$q(x+2 y+2 z-6)$

on comparing the coefficient;

$p+q=1$ and $p+2 q=3$

$\Rightarrow(\mathrm{p}, \mathrm{q})=(-1,2)$

Hence $x+3 y+\lambda z-\mu=x+3 y+3 z-7$

$\Rightarrow \lambda=3, \mu=7$