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The system of equations x + y + az = b, 2x + 3y = 2a, 3x + 4y + a²*z = ab + 2 has

1. unique solution when a != 0, b in R
2. no solution when a = 0, b = 1
3. infinite solutions when a = 0, b = 2
4. infinite solutions when a = 1, b in R

Correct answer: no solution when a = 0, b = 1

Solution

The determinant of the coefficient matrix is a(a-1), so unique solutions exist only when a != 0 AND a != 1. For a=0: equations give unique point x=6, y=-4, but this requires b=2 from eq(1); so a=0, b=1 gives no solution. For a=1: det=0; checking shows b must equal 1 for consistency, giving infinite solutions only when b=1 (not all b in R).

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