1. **State the problem:** We are given a trapezoid STVW with midsegment RU. The lengths are given as VW = $3y - 56$, RU = $-2y + 64$, and ST = $y$. We need to find the value of $y$. 2. **Recall the midsegment formula for trapezoids:** The midsegment (RU) is parallel to the bases (ST and VW) and its length is the average of the lengths of the two bases. $$RU = \frac{ST + VW}{2}$$ 3. **Substitute the given expressions:** $$-2y + 64 = \frac{y + (3y - 56)}{2}$$ 4. **Simplify the right side:** $$-2y + 64 = \frac{4y - 56}{2}$$ 5. **Simplify the fraction:** $$-2y + 64 = 2y - 28$$ 6. **Solve for $y$ by bringing all terms to one side:** $$-2y + 64 = 2y - 28$$ Add $2y$ to both sides: $$\cancel{-2y} + 64 + \cancel{2y} = 2y - 28 + 2y$$ $$64 = 4y - 28$$ Add 28 to both sides: $$64 + 28 = 4y - 28 + 28$$ $$92 = 4y$$ Divide both sides by 4: $$\frac{92}{\cancel{4}} = \frac{4y}{\cancel{4}}$$ $$23 = y$$ 7. **Final answer:** $$\boxed{23}$$