1. **State the problem:** We need to find the slope of line $k$ which is perpendicular to line $j$. Line $j$ passes through points $(1, 5)$ and $(7, 10)$. 2. **Formula for slope:** The slope $m$ of a line passing through points $(x_1, y_1)$ and $(x_2, y_2)$ is given by: $$m = \frac{y_2 - y_1}{x_2 - x_1}$$ 3. **Calculate slope of line $j$:** $$m_j = \frac{10 - 5}{7 - 1} = \frac{5}{6}$$ 4. **Rule for perpendicular slopes:** If two lines are perpendicular, their slopes are negative reciprocals of each other. That means: $$m_k = -\frac{1}{m_j}$$ 5. **Calculate slope of line $k$:** $$m_k = -\frac{1}{\frac{5}{6}} = -\frac{6}{5}$$ 6. **Final answer:** The slope of line $k$ is: $$\boxed{-\frac{6}{5}}$$