1. **State the problem:** Solve the equation $$\frac{3}{2}y + 25 = \frac{7}{3}y + 10$$ for $y$. 2. **Write down the equation:** $$\frac{3}{2}y + 25 = \frac{7}{3}y + 10$$ 3. **Isolate the variable terms on one side and constants on the other:** Subtract $\frac{7}{3}y$ from both sides and subtract 25 from both sides: $$\frac{3}{2}y - \frac{7}{3}y = 10 - 25$$ 4. **Find a common denominator to combine the $y$ terms:** The denominators are 2 and 3, so the common denominator is 6. $$\frac{3}{2}y = \frac{9}{6}y, \quad \frac{7}{3}y = \frac{14}{6}y$$ Rewrite the equation: $$\frac{9}{6}y - \frac{14}{6}y = -15$$ 5. **Combine the fractions:** $$\left(\frac{9}{6} - \frac{14}{6}\right) y = -15$$ $$\frac{9 - 14}{6} y = -15$$ $$\frac{-5}{6} y = -15$$ 6. **Solve for $y$ by dividing both sides by $\frac{-5}{6}$:** $$y = \frac{-15}{\frac{-5}{6}}$$ Rewrite division as multiplication by reciprocal: $$y = -15 \times \frac{6}{-5}$$ 7. **Simplify the expression:** $$y = -15 \times \frac{6}{-5} = \cancel{-15} \times \frac{6}{\cancel{-5}} = 3 \times 6 = 18$$ **Final answer:** $$y = 18$$