Line m passes through the origin. Line l is parallel to line m. What

Line m passes through the origin. Line l is parallel to line m. What are the equations of the two lines?

Equation of a line in point intercept form is \(y=mx+b\), where: \(m\) is the slope of the line and \(b\) is the y-intercept of the line (the value of \(y\) for \(x=0\)).

From the stem:
Since parallel lines have the same slope, then the slopes of l and m are the same;
Since a line passing through the origin has y-intercept equal to zero then the equation of line m would be \(y_m=mx\) and the equation of line l would be \(y_l=mx+b\)

(1) The horizontal distance between the two lines is 5 units --> basically we are told that the x-intercept of line l is either -5 or 5, so we know that line l passes either through the point (-5, 0) or (5, 0). Not sufficient.

(2) Line l has a y-intercept of 2.5 --> \(b=2.5\), so we know that line l passes through the point (0, 2.5). One point is not enough to determine (fix) a line. Not sufficient.

(1)+(2) Now, even take together we cannot determine whether line l passes through the point (-5, 0) or (5, 0). So, we would have two possible points of line l: (-5, 0) and (0, 2.5) OR (5, 0) and (0, 2.5), which means that we would have two possible equations of line l and m: \(y_l=0.5*x+2.5\) and \(y_m=0.5*x\) OR \(y_l=-0.5*x+2.5\) and \(y_m=-0.5*x\). Not sufficient.

Answer: E.

To demonstrate.
Case 1:
Case 2:
For more on this subject please check Coordinate Geometry chapter of Math Book: math-coordinate-geometry-87652.html

Hope it helps.