1. **State the problem:** Solve the system of equations using the elimination method: $$\begin{cases} 2x + 3y = 12 \\ 2x + 3y = 1 \end{cases}$$ 2. **Observe the system:** Both equations have the same left-hand side $2x + 3y$ but different right-hand sides (12 and 1). 3. **Apply elimination:** Subtract the second equation from the first: $$ (2x + 3y) - (2x + 3y) = 12 - 1 $$ which simplifies to $$ 0 = 11 $$ 4. **Interpret the result:** The statement $0 = 11$ is false, indicating the system has no solution. 5. **Conclusion:** The system is inconsistent and the lines represented by these equations are parallel and do not intersect. **Final answer:** No solution (the system is inconsistent).