If the system of equations α x + y + z = 5 , x + 2 y + 3 z = 4 , x + 3 y + 5 z = β . Has infinitely many solutions, then the ordered pair α , β is equal to

Step-by-step solution

We know that, for infinitely many solutions, Δ = 0 = Δ x = Δ y = Δ z So, Δ = α 1 1 1 2 3 1 3 5 = 0 ⇒ α 10 - 9 - 1 5 - 3 + 1 3 - 2 = 0 ⇒ α - 2 + 1 = 0 ⇒ α = 1 Now checking for Δ x = 5 1 1 4 2 3 β 3 5 = 0 ⇒ 5 10 - 9 - 1 20 - 3 β + 1 12 - 2 β ⇒ 5 - 20 + 3 β + 12 - 2 β ⇒ - 3 + β = 0 ⇒ β = 3 Hence α , β = 1 , 3