For real numbers α and β , consider the following system of linear equations: x + y - z = 2 , x + 2 y + α z = 1 and 2 x - y + z = β . If the system has infinite solutions, then α + β is equal to ______.

Step-by-step solution

Given system of linear equations are x + y - z = 2 , x + 2 y + α z = 1 , 2 x - y + z = β Using Cramer Rule, we can say for infinite solutions Δ = Δ 1 = Δ 2 = Δ 3 = 0 Here, Δ = 1 1 - 1 1 2 α 2 - 1 1 = 0 Apply R 1 ↔ R 1 + R 3 ⇒ 3 0 0 1 2 α 2 - 1 1 = 0 ⇒ 3 2 + α = 0 ⇒ α = - 2 And Δ 2 = 1 2 - 1 1 1 - 2 2 β 1 = 0 ⇒ 1 1 + 2 β - 2 1 + 4 - β - 2 = 0 ⇒ β - 7 = 0 ⇒ β = 7 ∴   α + &#946