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 _____________.

Solution

Maths sir · 4 steps

Infinitely many solutions means the third equation is built from the first two. Find how it is built.

1. The condition. Three equations have infinitely many solutions only when one of them adds nothing new. Here that means the third equation is an exact combination a(eq 1) + b(eq 2), right-hand side included. In determinant language, Δ = Δₓ = Δ_y = Δ_z = 0.

   a(2x + y - z) + b(x - y - z) = 3x + 3y + β z

2. Match x and y. From x: 2a + b = 3. From y: a - b = 3. Adding them gives 3a = 6, so a = 2 and b = -1.
3. Match z and constants. From z: β = -a - b = -2 + 1 = -1. From the right side: 3a + bα = 3, so 6 - α = 3 and α = 3. As a check, Δ = -3β - 3 = 0 at β = -1, and with β = -1, Δₓ = 6 - 2α = 0 at α = 3.
4. So. α + β - αβ = 3 + (-1) - (3)(-1) = 3 - 1 + 3 = 5.

The common slip: -1. Treating -αβ as -3 and forgetting that β = -1 is negative gives 3 - 1 - 3 = -1, not 5.

The idea to keep. Infinitely many solutions means one equation adds nothing new. Write it as a combination of the others, match every coefficient and the constant, and the unknowns fall out.

Answer from the NTA final answer key. Solution by PrepWiser.