If line l passes through point (m, n), is the slope of the

The slope is defined as the ratio of the "rise" divided by the "run" between two points on a line, or in other words, the ratio of the altitude change to the horizontal distance between any two points on the line. Given two points \((x_1,y_1)\) and \((x_2,y_2)\) on a line, the slope \(m\) of the line is: \(m=\frac{y_2-y_1}{x_2-x_1}\)


If line l passes through point (m,– n), is the slope of the line negative?

(1) The line passes through point (–m, n) --> \(slope=\frac{n-(-n)}{-m-m}=-\frac{n}{m}\), so the question becomes: is \(-\frac{n}{m}<0\)? or do \(m\) and \(n\) have the same sign, but we don't know that. Not sufficient.

(2) mn is negative --> \(m\) and \(n\) have the opposite signs --> point (m, -n) is either in I or in III quadrant, though as we have only one point lines passing through it can have negative as well as positive slope. Not sufficient.

(1)+(2) As from (2) \(m\) and \(n\) have the opposite signs the from (1) \(slope=-\frac{n}{m}>0\) and the answer to the question is NO. Sufficient.

Answer: C.

Without any algebra:

If line l passes through point (m,– n), is the slope of the line negative?

(1) The line passes through point (–m, n). Two cases:

A. If m and n are both positive then point (m, -n)=(positive, negative) is in IV quadrant and the second point (-m, n)=(negative, positive) is in II quadrant line passing these two points will have negative slope;

B. If m and n have the opposite signs, for example m positive and n negative, (m, -n)=(positive, positive) is in I quadrant and the second point (-m, n)=(negative, negative) is in III quadrant, line passing these two points will have positive slope (if it's vise-versa, meaning if m is negative and n positive, then we'll still have the same quadrants: (m, -n)=(negative, negative) is in III quadrant and the second point (-m, n)=(positive, positive) is in I quadrant, line passing these two points will have positive slope). Not sufficient.

(2) mn is negative --> m and n have the opposite signs --> point (m, -n) is either in I quadrant in case (m, -n)=(positive, positive) or in III quadrant in case (m, -n)=(negative, negative), though as we have only one point lines passing through it can have negative as well as positive slope. Not sufficient.

(1)+(2) As from (2) m and n have the opposite signs then we have the case B from (1), whihc means that the slope is positive. Sufficient.

Answer: C.

Check Coordinate Geometry chapter of Math Book for more: math-coordinate-geometry-87652.html