1. **State the problem:** We are given two triangles, one larger and one smaller, where the larger triangle is a dilation of the smaller triangle with scale factor $k=2.5$. We need to find the values of $x$ and $y$ given the side lengths and angle measures: - Larger triangle side: $21.5x + 4$ - Larger triangle angle: $(4y + 17)^\circ$ - Smaller triangle side: $13x - 5$ - Smaller triangle angle: $(6y + 7)^\circ$ 2. **Recall the dilation properties:** When one figure is a dilation of another with scale factor $k$, corresponding lengths are related by: $$\text{length}_{large} = k \times \text{length}_{small}$$ Corresponding angles remain equal: $$\text{angle}_{large} = \text{angle}_{small}$$ 3. **Set up equations for sides and angles:** For sides: $$21.5x + 4 = 2.5(13x - 5)$$ For angles: $$(4y + 17) = (6y + 7)$$ 4. **Solve the side length equation:** $$21.5x + 4 = 2.5(13x - 5)$$ $$21.5x + 4 = 32.5x - 12.5$$ Bring all terms to one side: $$21.5x + 4 - 32.5x + 12.5 = 0$$ $$\cancel{21.5x} + 4 - \cancel{32.5x} + 12.5 = 0$$ $$-11x + 16.5 = 0$$ $$-11x = -16.5$$ $$x = \frac{-16.5}{-11} = 1.5$$ 5. **Solve the angle equation:** $$(4y + 17) = (6y + 7)$$ Bring all terms to one side: $$4y + 17 - 6y - 7 = 0$$ $$-2y + 10 = 0$$ $$-2y = -10$$ $$y = \frac{-10}{-2} = 5$$ 6. **Final answer:** $$x = 1.5, \quad y = 5$$