1. **Problem statement:** In parallelogram ABCD, given that $ED = 7x - 13$ and $BD = 16x - 38$, find the length of diagonal $BD$. 2. **Key property:** In a parallelogram, the diagonals bisect each other. This means point $E$ is the midpoint of diagonal $BD$, so $ED = EB$. 3. **Set up the equation:** Since $E$ is midpoint of $BD$, $ED = EB = \frac{BD}{2}$. 4. **Express $ED$ and $BD$ in terms of $x$:** $$ED = 7x - 13$$ $$BD = 16x - 38$$ 5. **Use midpoint property:** $$7x - 13 = \frac{16x - 38}{2}$$ 6. **Multiply both sides by 2 to clear denominator:** $$2(7x - 13) = 16x - 38$$ $$14x - 26 = 16x - 38$$ 7. **Rearrange to isolate $x$:** $$14x - 26 = 16x - 38$$ $$14x - 16x = -38 + 26$$ $$\cancel{14x} - \cancel{16x} = -12$$ $$-2x = -12$$ 8. **Divide both sides by -2:** $$\frac{-2x}{\cancel{-2}} = \frac{-12}{\cancel{-2}}$$ $$x = 6$$ 9. **Find $BD$ by substituting $x=6$:** $$BD = 16(6) - 38 = 96 - 38 = 58$$ **Final answer:** $$\boxed{58}$$