Consider the system of equations: x + y + z = 3, x – y + 2z = 6 and x + y + α z = β

For what value of α and β the system has unique solution.

Concept:

Let us consider a system of equations in three variables:

a₁ × x + b₁ × y + c₁ × z = d₁

a₂ × x + b₂ × y + c₂ × z = d₂

a₃ × x + b₃ × y + c₃ × z = d₃

Then, 

By cramer’s rule:

I. If Δ ≠ 0, then the system of equation has unique solution and it is given by: 

II. If Δ = 0 and atleast one of the determinants Δ, Δ₁, Δ₂ and Δ₃ is non-zero, then the given system is inconsistent.

III. If Δ = 0 and Δ₁ =  Δ₂ =  Δ₃ = 0, then the system is consistent and has infinitely many solutions.

Calculation:

Given: x + y + z = 3, x – y + 2z = 6 and x + y + α z = β.

As we know that,

⇒ Δ = 2 – 2α, Δ₁ = 3β – 9α, Δ₂ =  3α - β and Δ₃ = 6 – 2β.

As we know that, for the given system of equation to have unique solution according to cramer’s rule:

 Δ ≠ 0

⇒ Δ = 2 – 2α ≠ 0 ⇒ α ≠ 1.