Mathematics · System of Linear Equations

Let α, β, and γ be real numbers. Consider the following system of linear equations: x + 3y + z = 52x + αz = 83x - y + βz = γ Match List-I with List-II: List-I: (P) If β = 12(3α - 2) and γ = 10, then system has (Q) If β = 12(3α - 2) and γ ≠ 10, then system has (R) If β ≠ 12(3α - 2), where α = 2 and γ = 10, then system has (S) If β = 12(3α - 2), where α = 2 and γ = 10, then system has List-II: (1) a unique solution (2) no solution (3) infinitely many solutions (4) x = 2, y = 1, z = 0 as a solution