1. **Problem 9:** Given points D, E, F are collinear with segments DE and EF such that $DE = 7x + 1$ and $EF = 4x - 3$, and the total segment $DF = 42$. Find $DE$. 2. **Formula:** By the Segment Addition Postulate, $DF = DE + EF$. 3. **Set up equation:** $$7x + 1 + 4x - 3 = 42$$ 4. **Simplify:** $$11x - 2 = 42$$ 5. **Solve for $x$:** $$11x = 42 + 2$$ $$11x = 44$$ $$x = \frac{44}{11}$$ $$x = 4$$ 6. **Find $DE$ by substituting $x=4$:** $$DE = 7(4) + 1 = 28 + 1 = 29$$ **Final answer:** $DE = 29$.