1. **State the problem:** Simplify the expression $$\frac{6x - y}{10xy} + \frac{1}{2x} - \frac{2y - 7x}{5xy}$$ and show it equals $$\frac{k}{y}$$ where $$k$$ is an integer. 2. **Identify the common denominator:** The denominators are $$10xy$$, $$2x$$, and $$5xy$$. The least common denominator (LCD) is $$10xy$$. 3. **Rewrite each term with the LCD:** - First term is already over $$10xy$$. - Second term: $$\frac{1}{2x} = \frac{1 \times 5y}{2x \times 5y} = \frac{5y}{10xy}$$. - Third term: $$\frac{2y - 7x}{5xy} = \frac{(2y - 7x) \times 2}{5xy \times 2} = \frac{2(2y - 7x)}{10xy}$$. 4. **Combine the terms:** $$\frac{6x - y}{10xy} + \frac{5y}{10xy} - \frac{2(2y - 7x)}{10xy} = \frac{6x - y + 5y - 2(2y - 7x)}{10xy}$$ 5. **Simplify the numerator:** $$6x - y + 5y - 2(2y - 7x) = 6x - y + 5y - 4y + 14x = (6x + 14x) + (-y + 5y - 4y) = 20x + 0 = 20x$$ 6. **Rewrite the expression:** $$\frac{20x}{10xy}$$ 7. **Simplify the fraction:** $$\frac{20x}{10xy} = \frac{\cancel{20}^2 \cancel{x}}{\cancel{10}^1 \cancel{x} y} = \frac{2}{y}$$ 8. **Conclusion:** The expression simplifies to $$\frac{2}{y}$$, so $$k = 2$$ which is an integer. **Final answer:** $$\boxed{\frac{2}{y}}$$