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For the linear system x1 + 2x2 + 3x3 = 6 x1 + 3x2 + 5x3 = 9 2x1 + 5x2 + ax3 = b if it is consistent and admits infinitely many solutions, then which statement must be true?

1. a = 8, and b may be any real number
2. b = 15, and a may be any real number
3. a belongs to R \ {8} and b belongs to R \ {15}
4. a = 8, b = 15

Correct answer: a = 8, and b may be any real number

Solution

The system must have a specific relationship between the coefficients for it to be consistent with infinitely many solutions. Setting a = 8 ensures that the third equation is a linear combination of the first two, while b can vary freely, allowing for infinite solutions.

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