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If for some α and β in R, the intersection of the following three planes x + 4y − 2z = 1 x + 7y − 5z = β x + 5y + αz = 5 is a line in R^3, then α + β is equal to

1. 0
2. −10
3. 10
4. 2

Correct answer: 10

Solution

For the intersection of three planes to form a line, the normal vectors of the planes must be coplanar, which means the determinant of the matrix formed by their coefficients must be zero. Solving the determinant condition leads to the relationship between α and β, resulting in α + β = 10.

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