Question 14

None

The system of equations has no solutions, indicating that the lines represented by the equations do not intersect at any point.

A) None

This option is correct because the two equations represent parallel lines. The first equation, x + 4y = 3, can be rearranged to y = -(1/4)x + 3/4, while the second equation, 2x + 8y = 4, simplifies to y = -(1/4)x + 1/4. Since they have the same slope but different y-intercepts, they will never intersect, resulting in no solutions.

B) One

This choice is incorrect as it suggests that the two lines intersect at a single point. For a system of equations to have one solution, the lines must have different slopes, indicating they cross each other. In this case, the lines are parallel and do not meet.

C) Two

This option is also incorrect because two distinct lines can only intersect at one point unless they are the same line. In this instance, the lines are parallel and thus do not intersect at all, ruling out the possibility of two solutions.

D) Infinitely many

This choice is incorrect as it implies that the two equations represent the same line, which would yield infinitely many solutions. However, since the lines are parallel with different intercepts, they do not coincide, confirming that there are no solutions instead of infinitely many.

Conclusion

The correct answer is "None" because the given equations represent parallel lines that do not intersect. All other options suggest the existence of intersections or coincidences, which is not the case here, confirming the absence of solutions.