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In the xy-coordinate system, do any points on line k lie in quadrant III?

You should just visualize this to solve.

(1) Line k has y-intercept 2 --> if the slope is negative or zero then K will not pass through the III quadrant but if it's positive then it will. Not sufficient.

(2) Line k has slope 2/3 --> any line with positive slope passes through the quadrants I and III. Sufficient.

Answer: B.

Check the image below:

Attachment:

graph (2).png [ 8.49 KiB | Viewed 7145 times ]

Red and blue lines do not pass through III quadrant and green one does. Notice that any line parallel to green line (so with a slope of 2/3) will also pass through III quadrant.

Some notes: 1. If the slope of line is negative, 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, line intersects the quadrant I too, 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 of either of axis crosses two quadrants.

4. If a line is horizontal the line has slope 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.

Dear Bunuel, I just want to clear my doubt. If the 2nd statement had been " Line K has negative slope" then we would have required both statements together to answer the question. correct?? Regards, Narenn
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Dear Bunuel, I just want to clear my doubt. If the 2nd statement had been " Line K has negative slope" then we would have required both statements together to answer the question. correct?? Regards, Narenn

Absolutely. Notice that in this case the answer to the question would be NO.
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Let me make clear: you are correct the answer would be C, but the answer to the question whether "any points on line k lie in quadrant III" would be NO.
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