1. **State the problem:** Given points P(3, 0), Q(-1, 6), R(k, -6), and S(4, -7), with PQ parallel to RS, find the value of $k$. 2. **Recall the formula for slope:** The slope $m$ between two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by $$m = \frac{y_2 - y_1}{x_2 - x_1}$$ Parallel lines have equal slopes. 3. **Calculate slope of PQ:** $$m_{PQ} = \frac{6 - 0}{-1 - 3} = \frac{6}{-4} = -\frac{3}{2}$$ 4. **Calculate slope of RS:** $$m_{RS} = \frac{-7 - (-6)}{4 - k} = \frac{-1}{4 - k}$$ 5. **Set slopes equal since PQ $\parallel$ RS:** $$-\frac{3}{2} = -\frac{1}{4 - k}$$ 6. **Solve for $k$:** Multiply both sides by $4 - k$: $$-\frac{3}{2} (4 - k) = -1$$ Multiply both sides by $-1$: $$\frac{3}{2} (4 - k) = 1$$ Multiply both sides by 2: $$3(4 - k) = 2$$ Expand: $$12 - 3k = 2$$ Subtract 12 from both sides: $$-3k = 2 - 12 = -10$$ Divide both sides by $-3$: $$k = \frac{-10}{-3} = \frac{10}{3}$$ **Final answer:** $$k = \frac{10}{3}$$