1. **Stating the problem:** Simplify or analyze the expression $$\frac{x}{y+x}$$ and understand the given equation $$\frac{x}{y+x} = \frac{1}{y}$$. 2. **Understanding the equation:** We have $$\frac{x}{y+x} = \frac{1}{y}$$. This means the fraction on the left equals the fraction on the right. 3. **Cross-multiply to solve for variables:** Multiply both sides by $$y(y+x)$$ to clear denominators: $$x \cdot y = 1 \cdot (y+x)$$ which simplifies to $$xy = y + x$$. 4. **Rearrange the equation:** Move all terms to one side: $$xy - y - x = 0$$ or $$y(x - 1) - x = 0$$. 5. **Solve for y:** $$y(x - 1) = x$$ $$y = \frac{x}{x - 1}$$. 6. **Check the expression for $$x^2$$:** The user wrote $$x^2 = \frac{x^2}{y} + \frac{x^2}{y}$$ which simplifies to $$x^2 = 2 \cdot \frac{x^2}{y}$$. 7. **Solve for y from this:** Divide both sides by $$x^2$$ (assuming $$x \neq 0$$): $$1 = \frac{2}{y}$$ which gives $$y = 2$$. 8. **Summary:** From the original equation, $$y = \frac{x}{x-1}$$, and from the second expression, $$y=2$$. These can be combined to find $$x$$ if needed. **Final answers:** $$y = \frac{x}{x-1}$$ and $$y = 2$$ from the given expressions.