If the system of linear equations x + y + z = 5 x + 2y + 2z = 6 x + 3y + λ = μ , (λ, μ ∈ R),has infinitely many solutions, then the value of λ + μ is:

Question

If the system of linear equations x + y + z = 5 x + 2y + 2z = 6 x + 3y + λ = μ , (λ, μ ∈ R),has infinitely many solutions, then the value of λ + μ is: (A) 12 (B) 10 (C) 9 (D) 7

Tardigrade Admin · Accepted Answer

x + 3y + λ z - μ = p (x + y + z -5) + q ( x + 2y + 2z - 6) on comparing the coefficient; p + q = 1 and p + 2 q = 3 ⇒ ( p , q) = ( - 1 , 2 ) Hence x + 3y + λ z - μ = x + 3y + 3z -7 ⇒ λ = 3 , μ = 7

Q. If the system of linear equations
$x + y + z = 5$
$x + 2 y + 2 z = 6$
$x + 3 y + λ = μ, (λ, μ \in R)$ ,has infinitely many solutions, then the value of $λ + μ$ is:

12

10

9

7

$x + 3 y + λ z - μ = p (x + y + z - 5) + q (x + 2 y + 2 z - 6)$
on comparing the coefficient;
$p + q = 1$ and $p + 2 q = 3$
$\Rightarrow (p, q) = (- 1, 2)$
Hence $x + 3 y + λ z - μ = x + 3 y + 3 z - 7$
$\Rightarrow λ = 3, μ = 7$

Q. If the system of linear equations x+y+z=5 x+2y+2z=6 x+3y+λ=μ,(λ,μ∈R),has infinitely many solutions, then the value of λ+μ is:

12

10

9

7

x+3y+λz−μ=p(x+y+z−5)+q(x+2y+2z−6) on comparing the coefficient; p+q=1 and p+2q=3 ⇒(p,q)=(−1,2) Hence x+3y+λz−μ=x+3y+3z−7 ⇒λ=3,μ=7

Q. If the system of linear equations
$x + y + z = 5$
$x + 2 y + 2 z = 6$
$x + 3 y + λ = μ, (λ, μ \in R)$ ,has infinitely many solutions, then the value of $λ + μ$ is:

$x + 3 y + λ z - μ = p (x + y + z - 5) + q (x + 2 y + 2 z - 6)$
on comparing the coefficient;
$p + q = 1$ and $p + 2 q = 3$
$\Rightarrow (p, q) = (- 1, 2)$
Hence $x + 3 y + λ z - μ = x + 3 y + 3 z - 7$
$\Rightarrow λ = 3, μ = 7$