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For the system x + y + z = 6, x + 2y + 3z = 14, and 2x + 5y + pz = q, which statement is correct?

1. infinitely many solutions when p = 8, q = 36
2. unique solution when p != 8, q != 36
3. no solution when p = 8, q != 36
4. at least one solution for q = 36 and every p in R

Correct answer: no solution when p = 8, q != 36

Solution

The determinant is zero only at p = 8. Then the system is consistent (infinitely many) if q = 36 but inconsistent (no solution) if q != 36; hence 'no solution when p = 8, q != 36' is correct.

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