If the system of linear equations \[ \begin{aligned} & x+y+z=5 \\ & x+2 y+2 z=6 \end{aligned} \] \(x+3 y+\lambda z=\mu\) (\(\lambda, \mu \in R\)), has infinitely many solutions then the value of \(\lambda+\mu\) is:

JEE Main · 2019 · 10 Apr 2019 · Shift 1 · Matrices

If the system of linear equations

x + y + z = 5 x + 2 y + 2 z = 6

x + 3 y + λ z = μ

(

λ, μ \in R

), has infinitely many solutions then the value of

λ + μ

is:

Answer: (C)

For infinitely many solutions the third equation must be a combination of the first two. Writing

x + 3 y + λ z - μ = a (x + y + z - 5) + b (x + 2 y + 2 z - 6)

and matching coefficients, the

x

-terms give

a + b = 1

and the

y

-terms give

a + 2 b = 3

, so

b = 2

and

a = - 1

. Then

λ = a + 2 b = 3

and

μ = 5 a + 6 b = 7

, so

λ + μ = 10

JEE Main · 2019 · 10 Apr 2019 · Shift 1 · Single correct option