The system of equations above has how many solutions? x+4y=3, 2x+8y=4

Rationale

None

The system of equations given represents two lines that are parallel and do not intersect, meaning there are no solutions to the system. When the equations are simplified, they reveal that they represent the same slope but different y-intercepts, confirming their parallel nature.

A (None) CORRECT

This choice is correct because the two equations, when examined, show that they are parallel lines. The first equation can be rewritten as \(y = -\frac{1}{4}x + \frac{3}{4}\), and the second as \(y = -\frac{1}{4}x + \frac{1}{2}\). Since they have the same slope but different y-intercepts, they never meet, confirming that there are no solutions.

B (One) YOUR ANSWER

This choice is incorrect because a system of equations can only have one solution if the lines intersect at exactly one point. In this case, the two lines do not intersect, hence there cannot be just one solution.

C (Two) YOUR ANSWER

This choice is incorrect because a system can only have two solutions if it consists of two different lines that intersect at two distinct points, which is impossible in a two-dimensional space. Since these lines are parallel, they cannot intersect at all.

D (Infinitely many) YOUR ANSWER

This choice is incorrect as it suggests that the equations represent the same line, which would yield infinitely many solutions. However, since the two equations have the same slope but different y-intercepts, they represent distinct parallel lines with no points of intersection.

Conclusion

In this system of equations, the lack of intersection between the two parallel lines leads to the conclusion that there are no solutions. The analysis of the slopes and y-intercepts highlights that while the lines maintain the same direction, their distinct intercepts ensure they never meet, confirming the absence of any solutions to the given system.