In the coordinate plane system, does line L pass through 3rd quadrant?

Given : L is a line.

1) A line with a positive slope will always pass through 3rd quadrant. e.g. Let our line L be y = mx + c. In 3rd quadrant, y < 0 and x < 0. Since, y < 0, we get (mx + c) < 0. Also, m > 0 and x < 0, so, mx < 0. For (mx + c) < 0, -infinity < c < mx. Therefore, a solution exists here, hence, line L can passes through 3rd quadrant.
I hope I did not complicate this much.

2) Using (y - y1) = ((y2 - y1)/(x2 - x1))(x - x1) we get L as 3x - 2y + 6 = 0 which passes through the point (-4, -3) which lies in 3rd quadrant.

Therefore, both statements are sufficient on their own.

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Can someone confirm whether on the GMAT, "line" is always meant as a straight-line, or whether the test will clarify? I think it's particularly relevant for this question, as line is a bit ambiguous in my mind.

From Official Guide: In geometry, the word “line” refers to a straight line that extends without end in both directions.

Bunuel - Thanks, that's helpful and a good definition for me to remember!

How is A sufficient? Can someone elaborate please?

Source: GMAT Math Book
Courtesy: Bunuel Thank you


1. If the slope of a line is negative, the line WILL intersect quadrants II and IV. X and Y intersects of the line with negative slope have the same sign. Therefore if X and Y intersects are positive, the line intersects quadrant I; if negative, quadrant III.

2. If the slope of line is positive, line WILL intersect quadrants I and III. Y and X intersects of the line with positive slope have opposite signs. Therefore if X intersect is negative, line intersects the quadrant II too, if positive quadrant IV.

3. Every line (but the one crosses origin OR parallel to X or Y axis OR X and Y axis themselves) crosses three quadrants. Only the line which crosses origin (0,0) OR is parallel to either of axis crosses only two quadrants.

4. If a line is horizontal it has a slope of 0, is parallel to X-axis and crosses quadrant I and II if the Y intersect is positive OR quadrants III and IV, if the Y intersect is negative. Equation of such line is y=b, where b is y intersect.

5. If a line is vertical, the slope is not defined, line is parallel to Y-axis and crosses quadrant I and IV, if the X intersect is positive and quadrant II and III, if the X intersect is negative. Equation of such line is x=a, where a is x-intercept.

Well, a line with a positive slope ALWAYS passes through 1st and 3rd quadrant (you can try to draw any line with positive slope and check).

However, if this line Also passes through the origin, then it will ONLY pass through 1st and 3rd quadrant, not through any other quadrant.
If this line does not pass through the origin, then it will also pass through either 2nd quadrant Or 4th quadrant.

Once again, a line with positive slope passing through origin will pass through 1st and 3rd quadrants Only
.
A line with positive slope that does not pass through origin will pass EITHER through 1st, 3rd and 2nd quadrants OR it will pass through 1st, 3rd and 4th quadrants.

Thanks for the explanation.

==> In the original condition, there are 2 variables (m and n from y=mx+n) and in order to match the number of variables to the number of equations, there must be 2 equations. Since there is 1 for con 1) and 1 for con 2), C is most likely to be the answer. However, con1)=con2), hence it is always yes and sufficient. Therefore, the answer is D.

Answer: D

If the line is perpendicular to the X axis, then the slope can be +Infinity, which is considered +ve IMO, Does this make A-option not sufficient? Please shed some light on this

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