1. **State the problem:** We are given the system of equations: $$2x + 3y = 5$$ $$4x + 6y = 10$$ We need to determine how many solutions this system has. 2. **Understand the system:** A system of linear equations can have: - No solution (parallel lines) - One solution (intersecting lines) - Infinitely many solutions (same line) 3. **Check if the equations are multiples:** Multiply the first equation by 2: $$2 \times (2x + 3y) = 2 \times 5$$ $$4x + 6y = 10$$ This is exactly the second equation. 4. **Interpretation:** Since the second equation is just a multiple of the first, both represent the same line. 5. **Conclusion:** The system has infinitely many solutions because both equations represent the same line. **Final answer:** Infinitely many solutions.