1. **State the problem:** Simplify the expression $$\frac{a}{2x+2y} = \frac{3x+3y - b}{3x+3y}$$ and understand the relationship between the terms. 2. **Rewrite the denominators:** Notice that $$2x+2y = 2(x+y)$$ and $$3x+3y = 3(x+y)$$. 3. **Express the equation with factored denominators:** $$\frac{a}{2(x+y)} = \frac{3(x+y) - b}{3(x+y)}$$ 4. **Simplify the right side:** $$\frac{3(x+y) - b}{3(x+y)} = 1 - \frac{b}{3(x+y)}$$ 5. **Rewrite the equation:** $$\frac{a}{2(x+y)} = 1 - \frac{b}{3(x+y)}$$ 6. **Bring all terms to a common denominator:** Multiply both sides by $$6(x+y)$$ (the least common multiple of denominators 2 and 3): $$6(x+y) \times \frac{a}{2(x+y)} = 6(x+y) \times \left(1 - \frac{b}{3(x+y)}\right)$$ 7. **Simplify both sides:** Left side: $$6(x+y) \times \frac{a}{2(x+y)} = 3a$$ Right side: $$6(x+y) \times 1 - 6(x+y) \times \frac{b}{3(x+y)} = 6(x+y) - 2b$$ 8. **Set the simplified expressions equal:** $$3a = 6(x+y) - 2b$$ 9. **Solve for $$a$$:** $$a = \frac{6(x+y) - 2b}{3} = 2(x+y) - \frac{2b}{3}$$ **Final answer:** $$a = 2(x+y) - \frac{2b}{3}$$