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I reduced this to reduced row echelon and got the last row as (0, 0 , K-51 |1) I set k-51=0 and got k=51, which is not the right answer. I did this because it requires me to find a value of K for which the system has no solution. What am I doing wrong?

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    Can you provide steps. So that we can identify your mistake if any. – 2017-01-29
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    So my line of thought is correct? – 2017-01-29

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On solving I get (0, 0 , K+51 |1). Maybe you have mistake in sign.

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An alternative and easy approach would be to use the determinant and set it equal to $0$. $$\begin{vmatrix} 1 & 1 & 3 \\ 1 & 2 & -5 \\ 7 & 16 & k \end{vmatrix}=0$$ You can solve this with a Laplace expansion and obtain: $$k+51=0$$ Giving you: $$\boxed{k=-51}$$