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: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.

Check this: in-the-xy-coordinate-system-do-any-points-on-line-k-lie-in-86074.html and this: math-coordinate-geometry-87652.html for more.

Hope it helps.
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