BTGmoderatorDC wrote:In the xy-plane, if line k has negative slope, is the y-intercept of line k positive?

(1) The x-intercept of line k is less than the y-intercept of line k.

(2) The slope of line k is less than -2.

OA C

Source: Veritas Prep

Say the equation of the line is y = mx + c, where m is the slope and c is the y-intercept of the line

Since it is given that line k has a negative slope, we have m < 0

Thus, we can write the equation as y =-|m|x + c

We have to determine whether c < 0.

Let's take each statement one by one.

(1) The x-intercept of line k is less than the y-intercept of line k.

Rewriting the equation y =-|m|x + c as y - c = -|m|x

=> [-1/|m|]*y + c/|m| = x

Thus, we have x-intercept of the line k = c/|m|

From the given information, we have c/|m| < c. This information is insufficient since c may or may not be positive. Insufficient.

(2) The slope of line k is less than -2.

=> -|m| < -2 => |m| > 2. Insufficient to determine wether c < 0.

(1) and (2) together

From (1), we have c/|m| < c and from (2), we have |m| > 2. Let's assume that y-intercept c is positive. Thus, we have

If c is positive, from the given inequality c/|m| < c, we cancel c; thus, we have 1/|m| < 1, which is possible since |m| > 2.

The y-intercept c cannot be negative. let's see how.

If c is negative, from the given inequality c/|m| < c, we have [-c/|m|] < -c. We can cancel c; thus, we have -1/|m| < -1, which is NOT possible since |m| > 2.

Thus, c > 0. Sufficient.

The correct answer: C

Hope this helps!

-Jay
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