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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, b = 15

Solution

Subtracting R1 and R2 from R3 gives (0, 0, a-8 | b-15). For the system to be consistent with infinitely many solutions, this row must vanish: a - 8 = 0 and b - 15 = 0, so a = 8 and b = 15.

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