For many, coordinate geometry is already a daunting concept. When a question dispenses with the graph all together, students can feel even more at a loss. If you fall into this group, do not despair. Here is a helpful guideline:

## Do Not Always Draw the Graph

This advice may seem counterintuitive. After all, the problem didn’t provide a graph. Wouldn’t the first step be to graph the problem out?

Many coordinate geometry concepts sans graph are testing your conceptual thinking. Take the follow problem:

1. Which of the following lines do not contain coordinate points that are both negative?

**Solution:**

The slope formula is important – if the question is explicitly asking for the slope. What is often more important is knowing that a line with a negative slope – from left to right – slopes downward. A positive slope, unsurprisingly, slopes upward.

Think of it this way – start at a negative x-coordinate (say -2) of a line. If you were to place a ball on the line would roll down the line as it moved into positive territory for the x-coordinate? If so the line is negative, if not the line is positive.

For this problem, we are looking for a line that does not pass through the third quadrant – the quadrant in which x and y are both negative. Graph the point (-2, -2). That’s in the third quadrant.

Now here’s the big conceptual part – any line that slopes upwards will always pass through Quadrant III. Graph it if you have to – or simply imagine a line of infinite length sloping upwards. Anyway you try to do so there will always be the Third Quadrant waiting to claim a part of your line.

Now, imagine a downward sloping line. Is it also crossing through the third quadrant? Well, move the entire line to the right. At a certain point, your line will no longer be in the Third Quadrant. As long as that line crosses the y-axis at a positive value, it will never cross through the Third Quadrant.

Now you only need to find two things: a line that has a positive y-intercept and a negative slope. And that is much better than having to graph every one of the equations in answer choices A – E!

Only answer (C) x + y = 2, which can be re-written as y = -x + 2, has a negative slope (-1) and positive y-intercept (+2).

## Takeaway

If a coordinate geometry question does not provide a graph, it is often testing conceptual thinking. So unless you are really desperate (which can happen on the GRE), and have time to spare, avoid graphing and think conceptually.

If you are unsure what to do, take a step back from the problem and ask yourself: will graphing this problem out take a long time? If the answer is yes, then there is very likely a much faster, no-graph approach.

sir, we can solve it by our school knowledge also. why are u going for such complex explanation. just assume any negative value for x. if we dont get negative value of y for any value of X then it will be the answer.

Hi Kamesh,

There are many ways to solve each problem on the GRE, and we try to show the most efficient towards the correct answer. In this case, if we test values without understanding the underlying concepts that are being tested, then this question will either take a long time or you run the risk of getting a wrong answer. If, for example, you test a negative value of x and find that y is also negative, then you immediately know that the answer choice is incorrect and you can eliminate it. However, what if you test a negative value of x and find that y is positive? Just because that point is positive doesn’t mean that there are NO y values that are negative. You would have to test many different values, and you still couldn’t be sure of the answer. While this method might help you to eliminate answers, it could also leave you with several different options without a clear way to distinguish between them.

Our method in this blog post, on the other hand, allows you to conceptualize the graph with an understanding of the properties of a line. It’s not actually a difficult method. We just need to recognize that a line with no coordinate points that are both negative must have a negative slope and positive y-intercept. This conceptual knowledge allows us to quickly scan the answer choices and know which one is correct without a doubt. That is a foolproof and quick way to test the answers. While it might be difficult to think this way at first, this strategy is a great way to quickly and efficiently answer these types of questions 🙂

Another way to solve it would be to think of substituting x and y with two negative numbers and see if the equation keeps being meaningful.

Indeed, choice (C) x+y = 2 cannot have both x and y negative because 2 is a positive number and the sum of two negative numbers is negative too. That’s why it is the answer!

Hi Chris,

If we’re sure we have a negative slope line, can’t the y-intercept be zero? Line y = mx, for all negative numbers m, wouldn’t go through Q III (since the axes are not in any quadrants): consider y = -x, for example.

Can you please clarify if in fact the condition for the y-intercept is *non-negative* instead of *positive*?

Thanks a lot and congrats for the many brilliant explanations 🙂

Hi Ana,

You are absolutely correct–the line y=-x would also fulfill the requirements of this question! I believe that it’s not mentioned in the explanation because all of the answer choices include a y-intercept 🙂

It’s great that you are thinking about these posts critically! Keep up the good work 🙂

Hi Magoosh, i dont undersatnd why (E) cant be the answer. I have understood that y intercept should be positive but the slope can be both postive and negative. For such cases, a line will never pass through 3rd quadrant and we can consider (E) to be true too.

FYI, i hope i am right in considering that y intercept = the distance from x=0 to the point on y axis.

i am sorrry. Edits to the above post.

Hi Magoosh, i dont undersatnd why (1), (4) cant be the answers. I have understood that y intercept should be positive but the slope can be both postive and negative. For such cases, a line will never pass through 3rd quadrant and we can consider (1), (4) to be true too.

FYI, i hope i am right in considering that y intercept = the distance from x=0 to the point on y axis.

Great question Neha. Here’s the issue: If a line passes through the third quadrant (the bottom left-side quadrant on a graph), then it will have two points that are negative. And this can

onlybe avoided if there is both a positive y-intercept and a negative slope.Why? Because if you have a positive slope, it will move upward from left to right… which is the same as moving downward form right to left. This will bring the line into the third quadrant as the y value goes down.

To give just one example, let’s say you have a y-intercept of 1, and a slope of 1. When y is at 0, the x value will move to -1, as you follow the line downward and to the left. Then, when y is at -1, the line will go down and to the left one more unit on the slope, bringing x to -2. This brings the line into the third quadrant, giving you two negative coordinates of -2,-1. Keep going down and to the left, and you’ll remain in the third quadrant and continue to have pairs of negative coordinates.

This will be the case anytime you have a positive y coordinate and positive slope. Eventually, as you follow the line down and to the left, the line will enter the third quadrant, and stay there for the rest of its downward movement.

Hi Neha,

See my response to your other comment. You must have a positive y-intercept and a negative slope in order to avoid the third quadrant, and thus avoid any points where both x and y are negative.

However, you are right about the location of the y-intercept. 🙂 The y-intercept the part of the y plane that intercepts with the 0 value for x. Similarly, at the x-intercept, y is 0.

I can’t understand this. Here only co-ordinate points are talked about. Why are we calculating slope and intercept?If , then why not E?The same reason , y intercept negative and slope positive.

Hi Niladri,

So, this is definitely just one approach to this problem. You definitely don’t need to think about slopes and y-intercepts, but it helps to visual the problem. Now, again we determined that we require (1) a positive y-intercept and (2) a negative slope. Let’s see if Answer (E) satisfies these two requirements. Let’s first rephrase (E):

x – y = 2 [Subtract “x” from both sides]

-y = -x + 2 [Multiply both sides by “-1”]

y = x – 2

Alright, so does this have a positive y-intercept? No. We see that it is negative: “y = x

– 2“. So, (E) is not a correct answer. Only (C) satisfies both requirements. Try each answer choice out. I hope this helps! 🙂How about just the fact that you can’t have 2 negative numbers add together to get a positive number, so of course it has to be C).

Hi Ian,

You are correct, but this logic doesn’t quite tell us the sign of the two variables. We know that we need a negative slope and a positive y-intercept. If we look at (C), we see that there are two variables that equal a positive number. If we put aside algebra for a moment and just think about integer properties, we know that both x and y can’t be negative as you said. However, we don’t know if they are both positive or if one is positive and one is negative. Either could be the case, so this method doesn’t narrow down our answer enough to be sure about it. The best way to figure this out is to put the equation into slope-intercept form (easily done by subtracting y from both sides) to easily see that (C) meets both of our requirements.

Brilliant Approach for such a problem.Thanks Chris 🙂

On Chris’s behalf, you’re very welcome 😀

This was a bit confusing. The grammatical errors made it hard to understand and also there was not an explanation about why you couldn’t have two positive coordinates. Here’s the part that was difficult to understand, “If you were to place a ball on the line would roll down the line as it move into positive territory for the x-coordinate?”

So, basically the main question I am asking is, if the question is just asking for coordinates that are not both negative, why does that exclude an answer with coordinates that are both positive?

Thanks!

This is a tricky conceptual post, Jaime. I’ll be happy to provide some clarification and additional explanation.

First, I should clarify that we’re not excluding answers where both coordinates are positive. We’re excluding answers where both the

yintercept and the slope of the line are positive. In the correct answer, theyintercept must be positive, but the slope must be negative. (Theyintercept is the coordinate for y when x is 0, and the slope is the angle of a line as it moves across thexyplane.)First, let’s look at this statement:

To make a long story short, this means that the line will be higher on the left, and slant downward as it moves to the right. A line like this that moves downward from left to right on the coordinate plane is said to have a

negative slope.Now, a negative sloped line is the only kind of line that might never have two negative coordinates. On a negative sloped line, if there is a positive value for

yat theyintercept (the point on the line at whichxis 0), thenywill never have a negative value at any point on the line. This is because all negative values foryare in the third quadrant— the lower left-hand corner of thexyplane, where both coordinates are always negative. So right there, we know that theycoordinate must have a positive value. If you are still having trouble picturing this, here is a drawing: (Direct link to the image here: https://s31.postimg.org/6i4xejlaz/Corrdinate_Plane.png )As you can see, the blue line on the image will never have a negative value for both

xandyat the same time, because it will never pass through the third quadrant. As I mentioned, this is the kind of line we need– a negative sloped line with a positive value foryat theyintercept. In order foryto be positive at the y intercept,ywill need to equal the value forxplus a positive number in the equation for the line. Moreover, any line that slopes downward from left to right must multiplyxby a negative number in the equation for the line.This brings us to our answer. The only equation where

y= a negative form ofxplus a positive number is (C)x+y= 2, because answer (C) cab be rewritten asy= –x+ 2.Does this make sense? If you still have any doubts, let me know.

I understood a total of none of this. I am starting to think the only math I will never understand is coordinate geometry. I thought I was supposed to plug in zero to solve for each and none I tested had two negative coordinates….I have no idea what anyone is talking about. 🙁

Hi Lauren,

I’m sorry this doesn’t make sense to you! I’ve sent your question to our test prep experts and they will respond to you by email soon. 🙂

thanks chris.

Dear Chris,

may be i was wrong or not got the point ! My query was a line with negative slope and positive y intercept may has coordinate point at quadrant 3 where both x and ya are negative. In addition , a line with positive slope with negative y intercept could also passes through the quadrant 3. I am screwed up and please help me out. 🙁

Good One! 🙂 cleared a lot of things for me

Chirs. This post is really helpful. Thanks a lot. 🙂

You are GRE guru. 😀

You are welcome!

Also one more point to notice : if both x-intercept and y-intercept of the line have same sign then slope is negative… else slope is positive

Great post! Thanks so much!

You are welcome!

in fact you do employ graphing here, but only this happens at mental state. You do graph to see that slopes of negative and positive will show up in certain quadrants or you imagine graph in your mind and then you isolate the line from quadrant IV to help the condition take effect. The only way is to attach positive y-intercept of course. I personally think sensible application of graphing for coordinate geometry saves time and efforts

excuse me, I meant isolate the line from quadrant III actually

Sure, I think it is a question of ‘sensible’. In this case, you could graph each line and definitely get the answer (2 mins) or think about it intuitively (30 – 45 sec.) Again, it depends on the person, but I would encourage others, at least while they are practicing, to try the intuitive approach as well :).

Nice post

You are welcome!