The system of equations x + 2y = 13 and 3x + 6y = 9 has :

Q: The system of equations x + 2y = 13 and 3x + 6y = 9 has :

The given system of equations is: 1. $ x + 2y = 13 $ 2. $ 3x + 6y = 9 $ To determine the nature of the solutions for this system, we can analyze the equations. ### Explanation for Option 2: No Solution 1. **Proportionality Check**: - The second equation $ 3x + 6y = 9 $ can be rewritten by dividing the entire equation by 3, resulting in $ x + 2y = 3 $. - Now, compare this with the first equation $ x + 2y = 13 $. 2. **Contradiction**: - Both equations have the same left-hand side, $ x + 2y $, but different right-hand sides (13 and 3). This indicates a contradiction because the same expression cannot equal two different values simultaneously. 3. **Conclusion**: - Since the two equations represent parallel lines (same slope but different y-intercepts), they do not intersect at any point on the Cartesian plane. Therefore, there is no solution to this system of equations. ### Explanation for Why Other Options are Incorrect 1. **Option 1: Unique Solution**: - A unique solution exists when two lines intersect at exactly one point. However, as shown above, the lines are parallel and do not intersect, so there is no unique solution. 2. **Option 3: Infinitely Many Solutions**: - Infinitely many solutions occur when the two equations represent the same line, meaning they are identical. In this case, the equations are not identical due to the different constants (13 and 3), so there cannot be infinitely many solutions. 3. **Option 4: Finite Number of Solutions**: - A finite number of solutions other than one implies multiple distinct intersection points, which is not possible for linear equations in two variables. Since the lines are parallel and do not intersect, there are no solutions, finite or otherwise. In conclusion, the correct answer is Option 2: No solution, as the system of equations represents parallel lines that do not intersect.

Question

Accepted Answer

The given system of equations is:

1. $ x + 2y = 13 $
2. $ 3x + 6y = 9 $

To determine the nature of the solutions for this system, we can analyze the equations.

### Explanation for Option 2: No Solution

1. **Proportionality Check**:
   - The second equation $ 3x + 6y = 9 $ can be rewritten by dividing the entire equation by 3, resulting in $ x + 2y = 3 $.
   - Now, compare this with the first equation $ x + 2y = 13 $.

2. **Contradiction**:
   - Both equations have the same left-hand side, $ x + 2y $, but different right-hand sides (13 and 3). This indicates a contradiction because the same expression cannot equal two different values simultaneously.

3. **Conclusion**:
   - Since the two equations represent parallel lines (same slope but different y-intercepts), they do not intersect at any point on the Cartesian plane. Therefore, there is no solution to this system of equations.

### Explanation for Why Other Options are Incorrect

1. **Option 1: Unique Solution**:
   - A unique solution exists when two lines intersect at exactly one point. However, as shown above, the lines are parallel and do not intersect, so there is no unique solution.

2. **Option 3: Infinitely Many Solutions**:
   - Infinitely many solutions occur when the two equations represent the same line, meaning they are identical. In this case, the equations are not identical due to the different constants (13 and 3), so there cannot be infinitely many solutions.

3. **Option 4: Finite Number of Solutions**:
   - A finite number of solutions other than one implies multiple distinct intersection points, which is not possible for linear equations in two variables. Since the lines are parallel and do not intersect, there are no solutions, finite or otherwise.

In conclusion, the correct answer is Option 2: No solution, as the system of equations represents parallel lines that do not intersect.