“Line K in Quadrant III”: Is there an easier way to solve problems such as this?

Question

Is there an easier way to solve problems such as this?

I mean we know that quadrants have their respective x, y properties, Q1 +x,+y, Q2 -x,+y, etc.

Is there a counterpart for determining such in a slope?

Say for instance m = 2/3 will never pass Q4. In general, if slope is positive, it never passes Q4. Similarly, if slope is negative, it never passes Q3. Is this true?

Answer

Yes, that's definitely an easier way to solve this. If a line has a positive slope AND a positive y-intercept, then it will never pass through Q4. You can come up with similar rules for each combination:

• positive slope, negative y-intercept - never pass through Q2
• negative slope, positive y-intercept - never pass through Q3
• negative slope, negative y-intercept - never pass through Q1

We can also state these rules:

• positive slope - will always pass through Q1 and Q3
• negative slope - will always pass through Q2 and Q4

From Statement 2, we know that the slope is positive. Therefore, we know that it will pass through Q3, so it is sufficient.

Link

http://gmat.magoosh.com/questions/1028