Breaking Down a GMATPrep Coordinate Plane Problem
This week, were going to tackle a harder GMATPrep problem solving question from the topic of Coordinate Planes (a subset of Geometry).
Lets start with the problem. Set your timer for 2 minutes. and GO!
*In the xy-plane, does the line with equation y=3x+2 contain the point (r, s)?(1) (3r + 2 s)(4r + 9 s) = 0
(2) (4r 6 s)(3r + 2 s) = 0
The problem gives us a formula; thats probably the first thing that most people will write down. The problem also mentions that this formula is for a line in the coordinate plane; some people may also draw a coordinate plane and sketch the line. (Its up to you whether to do this right away or whether to decide later whether such a drawing will be helpful.)
The problem wants to know whether the line contains the point (r, s) a specific point, but defined only in terms of variables. Thats kind of annoying. Can I just pick some numbers here?
Maybe. I have to be careful though. How would I decide what to pick for r and s? I cant pick based upon the equation given in the question, because that equation is part of the question I dont know whether that information is true. Id have to pick instead based upon the statements, which I know are true. As it turns out, if I do that, I end up just doing a longer version of a theoretical approach to the problem. In practice, I might try some numbers first to understand whats going on, but Id then want to learn to streamline my work so that I dont need to do that on the real test.
Given this equation from statement 1:
(3r + 2 s)(4r + 9 s) = 0
What do we know to be true? Well, these two quantities, (3r + 2 s) and (4r + 9 s), multiply to zero. In order for that to happen, it must be true that at least one of those expressions equals zero. (Possibly both equal zero, but that doesnt have to be true. Its enough for just one expression to equal zero.)
So, at the least, we know that either (3r + 2 - s)=0 or (4r + 9 - s)=0. How does that help?
Hmm. Im not quite sure how to connect that to the question. Maybe I have a little more work to do to rephrase the question. (Do you remember how to rephrase DS questions? If not, click here to learn how.)
The question is asking me whether the point (r, s) is included in the line y=3x+2. Thats the same thing as saying: is it true that s = 3r + 2? Notice that all I did was plug the point (r, s) into the equation. If point (r, s) is included in this line, then the equation should be true.
So heres my new question: is it true that s = 3r + 2?
That new equation looks a little bit more like the info I got out of statement one. Lets rearrange to look more like my question:
either (3r + 2 s)=0 or (4r + 9 s)=0
either s = 3r + 2 or s = 4r + 9
If the first equation is true, then the answer to my question is a definite yes. If, on the other hand my second equation is the one thats true, I have a definite I dont know. Maybe. Maybe not. Either equation might be the true one, so I dont have a definitive answer here. Statement 1 is insufficient. Cross off answers A and D.
Now try statement 2:
(4r 6 s)(3r + 2 s) = 0
Do the same thing!
either (4r 6 s)=0 or (3r + 2 s)=0
either s = 4r 6 or s = 3r + 2
Hmm. Im getting the same response. This time, if the second equation is the true one, then I get a definite yes answer but the first equation gives me a maybe answer. This statement is also insufficient. Cross off answer B.
Now I need to look at the two together. The info above all stays the same, but now the two statements can be applied at the same time.
I still dont have any more info about which equation in each pair might be the true one, though, so it looks like this one cant be done. Right?
Not so fast! Lets recap what we know.
either s = 3r + 2 is true or s = 4r + 9 is true
and, at the same time
either s = 4r 6 is true or s = 3r + 2 is true
The key is the at the same time part. Match up the equations. The equation s = 3r + 2 shows up in both halves, so if one of those two equations is true, then of course the other is true, too theyre the same equation. If the equation is true for one statement, then, that same equation must also be true for the other statement.
What about the other two though? Can s = 4r + 9 and s = 4r 6 both be true at the same time?
No. 4r = 4r, so thats the same in both equations. In one equation we add 9 to 4r and in the other equation we subtract 6 from 4r. It doesnt matter what 4r is you cant add 9 and subtract 6 from the exact same number and get the same answer. Impossible! So those two equations cannot be true together, which leaves us saying that the other two equations (really just one) must be true.
And, as we already discovered earlier, if the equation s = 3r + 2 is true, then we have a definitive yes answer for this yes/no data sufficiency question.
The correct answer is C.
Key Takeaways for Solving Hard Coordinate Plane Problems:
- Know that, given an equation and a particular point, that point can be substituted into the equation. In this case, this information was given in the question, but rephrasing the equation in terms of r and s made it easier to assess the statements.
- Know what the GMAT is really trying to tell you when you see anything in the form of (something) * (something else) = 0. That always means that at least one of those terms must be zero and thats very useful information to know!
- If you are a more visual thinker, draw pictures whenever you can. In this case, sketching the line on a coordinate plane and then thinking about individual points for (r, s) might have helped you to realize that you could rewrite the given equation (which came in terms of x and y) using r and s instead.
* GMATPrep questions courtesy of the Graduate Management Admissions Council. Usage of this question does not imply endorsement by GMAC.