1. **State the problem:** Solve the system of equations by substitution method: $$x + y = 2$$ $$2x^2 + xy + 1 = 0$$ 2. **Express one variable in terms of the other:** From the first equation, solve for $y$: $$y = 2 - x$$ 3. **Substitute into the second equation:** Replace $y$ with $2 - x$ in the second equation: $$2x^2 + x(2 - x) + 1 = 0$$ 4. **Simplify the equation:** $$2x^2 + 2x - x^2 + 1 = 0$$ $$x^2 + 2x + 1 = 0$$ 5. **Factor the quadratic:** $$(x + 1)^2 = 0$$ 6. **Solve for $x$:** $$x = -1$$ 7. **Find $y$ using $y = 2 - x$:** $$y = 2 - (-1) = 3$$ **Final solution:** $$\boxed{(x, y) = (-1, 3)}$$