1. **State the problem:** We are given two sets of points in the plane: $$A = \{(x,y) : y = \frac{1}{x}, x \neq 0, x \in \mathbb{R}\}$$ $$B = \{(x,y) : y = -x, x \in \mathbb{R}\}$$ We need to find the intersection $A \cap B$, i.e., points $(x,y)$ that satisfy both equations simultaneously. 2. **Set the equations equal:** Since points in the intersection must satisfy both equations, we set their $y$ values equal: $$\frac{1}{x} = -x$$ 3. **Solve for $x$:** Multiply both sides by $x$ (not zero): $$1 = -x^2$$ 4. **Analyze the equation:** $$x^2 = -1$$ Since $x^2$ is always non-negative for real $x$, and $-1$ is negative, there is no real solution. 5. **Conclusion:** There are no points $(x,y)$ in $\mathbb{R}^2$ that satisfy both equations simultaneously. Therefore, the intersection $A \cap B$ is the empty set $\emptyset$. **Final answer:** $$A \cap B = \emptyset$$