A line passing through $(-5, k)$ and $(13,-7)$ has a slope of $-\frac{1}{2}$. What is the value of $k$ ?

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The slope of the line is $\frac{-7-k}{13-(-5)}=-\frac{7+k}{18}$.

Therefore, $-\frac{7+k}{18}=-\frac{1}{2}=-\frac{9}{18}$, so $7+k=9$ from which it follows that $k=2$.