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For the following linear system, determine when it has no solution: 3x - 2y - kz = 10, 2x - 4y - 2z = 6, and x + 2y - z = 5m.

1. k = 3, m = 4/5
2. k ≠ 3, m is any real number
3. k ≠ 3, m ≠ 4/5
4. k = 3, m ≠ 4/5

Correct answer: k = 3, m ≠ 4/5

Solution

Determinant = 24 - 8k, which is 0 only at k=3. With k=3 the equations force 4y=1 from eqs 1&2 but 4y=5m-3 from eqs 2&3, consistent only if 5m-3=1 i.e. m=4/5. So no solution when k=3 and m != 4/5.

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