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The slope of a line passing through two points $(x_1,y_1)$ and $(x_2,y_2)$ is given by the formula $\frac{y_2-y_1}{x_2-x_1}$.
Using this formula, we have:
slope $=\frac{1-5}{3-6}=\frac{-4}{-3}=\boxed{\textbf{(C)}\ \frac{4}{3}}$