Question

If the system of linear equations 

2x + y – z = 3 

x – y – z = α

3x + 3y + βz = 3 

has infinitely many solution, then α + β – αβ is equal to _________.

Answer 1

2x + y – z = 3 ...(i)

x – y – z = α ...(ii)

3x + 3y + βz = 3 ...(iii)

2 × (i) – (ii) – (iii) gives :

-1(1+β)z = 3 - α

For infinitely many solution,

β+1 = 0 = 3 - α

⇒ (α,β) = (3,-1)

Hence,

α + β - αβ = 5