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An ordered pair (α, β) for which the system of linear equations (1 + α)x + βy + z = 2 αx + (1 + β)y + z = 3 αx + βy + 2z = 2 has a unique solution is:

1. (-3, 1)
2. (1, -3)
3. (-4, 2)
4. (2, 4)

Correct answer: (2, 4)

Solution

The ordered pair (2, 4) ensures that the determinant of the coefficient matrix of the system is non-zero, which is a requirement for the system to have a unique solution. This means that the equations are linearly independent and intersect at a single point.

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