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The system of linear equations x+y+z=6 2 x+5 y+a z=36 x+2 y+3 z=b has

Options

  1. Aunique solution for a=8 and b=16
  2. Binfinitely many solutions for a=8 and b=16
  3. Cunique solution for a=8 and b=14
  4. Dinfinitely many solutions for a=8 and b=14

Correct answer

D. infinitely many solutions for a=8 and b=14

Step-by-step solution

Coefficient matrix determinant: = 1(15-2a) - 1(6-a) + 1(4-5) = 8 - a . For a = 8 , = 0 , so unique solution is not possible. With a = 8 : R₂ - 2R₁ gives 3y + 6z = 24 y + 2z = 8 . R₃ - R₁ gives y + 2z = b - 6 . For consistency: b - 6 = 8 b = 14 . When a = 8, b = 14 : the system has infinitely many solutions.

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