Let the system of equations x + 2 y + 3 z = 5 , 2 x + 3 y + z = 9 , 4 x + 3 y + λ z = μ have infinite number of solutions. Then λ + 2 μ is equal to:

Step-by-step solution

Given: x + 2 y + 3 z = 5 , 2 x + 3 y + z = 9 and 4 x + 3 y + λ z = μ . For infinite solutions, Δ = Δ 1 = Δ 2 = Δ 3 = 0 . ⇒ Δ = 1 2 3 2 3 1 4 3 λ = 0 ⇒ 3 λ - 3 - 4 λ + 8 + 18 - 36 = 0 ⇒ λ = − 13 Now, finding ⇒ Δ 1 = 5 2 3 9 3 1 μ 3 − 13 = 0 ⇒ μ = 15 Now, finding ⇒ Δ 2 = 1 5 3 2 9 1 4 μ λ ⇒ Δ 2 = 1 5 3 2 9 1 4 15 − 13 ⇒ Δ 2 = 0 And ⇒ Δ 3 = 1 2 5 2 3 9 4 3 μ ⇒ Δ 3 = 1 2 5 2 3 9 4 3 15 ⇒ Δ 3 = 0 For λ = − 13 , μ = 15 system of equation has infinite solutions. ⇒ λ + 2 μ = 17 .