The following is a great coordinate geometry DS question that has been floating around with some of my students.

In the xy-plane, region R consists of all the points (x,y) such that 2x+3y<=6. Is the point (r,s) in region R?

(1) 3r+2s=6
(2) r<=3 and s<=2

It is a bit tough to explain in writing without being able to draw a picture, but if you could to that to follow along, would probably be a bit easier to understand if you draw this out on paper. And being a coordinate geometry question, I would recommend that you ALWAYS draw it out.

So we need to draw out this line and area of what R could be. Thus we have an area of 2x+3y<=6, so this is a line that goes through (3,0) and (0,2) So anything on this line and to the left of this line includes area R.

Now from 1), we must draw this line that is given. 3r+2s = 6. So we pick two easy points and see that the line goes through (2,0) and (0,3). As you can see, some points on this line are in the region R, and some are not, so insufficient on its own.

From 2), we are given some points that r is less than or equal to 3 and s is less than or equal to 2. Thus, we can see that some points could lie in the region R and some points could fall out of it, so not sufficient.

Combining, we still cannot determine, as there is some overlap of the line and the points that are outside of R, while there are some points that fall on the line, within region R and within the confines of statement 2. For example, (1,2) is on the line and fits within statement 2), but not within R. The point (2,0) is also on the line in 1), within R, and also within the region of statement 2). So we can see that when combining them, they are still insufficient.