1. **State the problem:** Solve the equation $\tan(xy) = e^{xy} + x$ for variables $x$ and $y$. 2. **Understand the equation:** The equation involves a transcendental function $\tan(xy)$ and an exponential function $e^{xy}$ plus a linear term $x$. 3. **Rewrite the equation:** Let $z = xy$. Then the equation becomes: $$\tan(z) = e^{z} + x$$ 4. **Analyze the equation:** Since $z = xy$, $y = \frac{z}{x}$ (assuming $x \neq 0$). 5. **Express $y$ in terms of $x$ and $z$:** $$y = \frac{z}{x}$$ 6. **Rewrite the original equation in terms of $x$ and $z$:** $$\tan(z) = e^{z} + x$$ 7. **Isolate $x$:** $$x = \tan(z) - e^{z}$$ 8. **Substitute back to find $y$:** $$y = \frac{z}{x} = \frac{z}{\tan(z) - e^{z}}$$ 9. **Summary:** The solutions $(x,y)$ satisfy: $$x = \tan(z) - e^{z}, \quad y = \frac{z}{\tan(z) - e^{z}}$$ where $z$ is any real number such that the denominator is not zero and $x \neq 0$. This implicit form expresses $x$ and $y$ parametrically in terms of $z$. **Note:** There is no closed-form explicit solution for $y$ solely in terms of $x$ due to the transcendental nature of the equation.