1. Stating the problem: Solve the equation $$e^{x+y} - 3xy - 2 = y$$ for the relationship between $$x$$ and $$y$$. 2. Rearrange the equation to isolate terms: $$e^{x+y} - 3xy - 2 = y \implies e^{x+y} - 3xy - 2 - y = 0$$ 3. Group terms for clarity: $$e^{x+y} - 3xy - y - 2 = 0$$ 4. Since the equation is transcendental (includes both exponential and polynomial terms), express $$y$$ implicitly as a function of $$x$$ or vice versa. Explicit closed-form solutions may not be straightforward. 5. Alternatively, rewrite as: $$e^{x+y} = 3xy + y + 2 = y(3x + 1) + 2$$ 6. This equation defines an implicit relation between $$x$$ and $$y$$: $$e^{x+y} = y(3x + 1) + 2$$ This relation can be analyzed graphically or numerically to find values of $$x$$ and $$y$$ satisfying the equation. Final answer: The equation $$e^{x+y} = y(3x + 1) + 2$$ implicitly defines $$y$$ in terms of $$x$$.