1. **State the problem:** Solve the system of equations: $$x - y = 2$$ $$x^2 + y^2 = 2$$ 2. **Use substitution:** From the first equation, express $x$ in terms of $y$: $$x = y + 2$$ 3. **Substitute into the second equation:** Replace $x$ with $y + 2$: $$ (y + 2)^2 + y^2 = 2 $$ 4. **Expand and simplify:** $$ y^2 + 4y + 4 + y^2 = 2 $$ $$ 2y^2 + 4y + 4 = 2 $$ 5. **Bring all terms to one side:** $$ 2y^2 + 4y + 4 - 2 = 0 $$ $$ 2y^2 + 4y + 2 = 0 $$ 6. **Divide entire equation by 2 to simplify:** $$ \cancel{2}y^2 + \cancel{2} \cdot 2 y + \cancel{2} = 0 \Rightarrow y^2 + 2y + 1 = 0 $$ 7. **Recognize perfect square:** $$ (y + 1)^2 = 0 $$ 8. **Solve for $y$:** $$ y + 1 = 0 \Rightarrow y = -1 $$ 9. **Find $x$ using $x = y + 2$:** $$ x = -1 + 2 = 1 $$ 10. **Final solution:** $$ (x, y) = (1, -1) $$ This is the only solution where the line and the circle intersect.