1. **State the problem:** We have a rectangle QRST with sides QR = $x$, QT = $3x - 10$, RS = $2y$, and TS = $4y$. Since QRST is a rectangle, opposite sides are equal in length. 2. **Set up equations:** Opposite sides are equal, so: $$QR = TS \implies x = 4y$$ $$QT = RS \implies 3x - 10 = 2y$$ 3. **Substitute $x = 4y$ into the second equation:** $$3(4y) - 10 = 2y$$ $$12y - 10 = 2y$$ 4. **Solve for $y$:** $$12y - 10 = 2y$$ $$12y - \cancel{10} = 2y + \cancel{10}$$ $$12y - 2y = 10$$ $$10y = 10$$ $$y = \frac{10}{10} = 1$$ 5. **Find $x$ using $x = 4y$:** $$x = 4(1) = 4$$ 6. **Calculate side lengths:** $$QR = x = 4$$ $$QT = 3x - 10 = 3(4) - 10 = 12 - 10 = 2$$ $$RS = 2y = 2(1) = 2$$ $$TS = 4y = 4(1) = 4$$ **Final answer:** $$x = 4, y = 1$$ Side lengths: QR = 4, QT = 2, RS = 2, TS = 4