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1. If the slope of a line is negative, the 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, the line intersects quadrant I; 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 OR is parallel to either of axis crosses only two quadrants.

4. If a line is horizontal it has a slope of , 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 , where a is x-intercept.

Hi Bunuel, Small corrections here- some of the words need to be intercepts and not intersect.

echoing 'MateoLibre' - it is indeed a v helpful post. I have a DS Q....(source: GMATPrep CAT)

In the xy plane at what 2 points does the graph of y = (x+a)(x+b) intersect the x axis?

1] a+b = -1 2] the graph intersects the y axis at (0,-6)

ANY help on how to solve this would be much appreciated. thanks.

This one is a very interesting problem, let us approach it in steps.

you are given y = (x+a)(x+b) and asked when this will intersect the x axis. Anytime you see "intersect the x axis", you know that y=0 So set y=0 and solve (x+a)(x+b)=0 you get x^2 + x (a+b) + ab = 0

Now look at the statements... 1) gives you a+b, but we don't know what ab is...INSUFF 2) gives you "intersects the y axis @ (0,-6).

So in original statement: y = (x+a)(x+b) ---> plug in (0,-6) You will get ab value. but this is INSUFF because we don't know a+b

So, answer is C since you need both 1 and 2 to solve.

When I have two points in a coordinate system e.g. (2,3) and (6,7) that pass through a line how do I know which number is x1 and which is x2 when calculating slope. Also, the same for the y1 and y2?

It doesn't matter: \(slope=\frac{7-3}{6-2}=\frac{3-7}{2-6}=1\).
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To answer, we must find the slope of each line and then check to see if one slope is the negative reciprocal of the other or if their product equals to -1. SlopeAB=5−199−48=−14−39=0.358SlopeAB=5−199−48=−14−39=0.358

SlopeCD=24−422−31=20−9=−2.22

The formula of the slope of two given coordinates are y2-y1 / x2-x1

However in some questions, 2nd coordinates (x2 y2)s are subtracted from 1sts (x1 y1) and in some, other way around. Can you please clarify what do we take into account concerning this formula?

Hi,

This is a mathematical rule. You can write a-b = -(b-a). Did you get this rule??

Now, in a similar way if you take - sign common from both numerator and denominator of y2-y1 / x2-x1,

you will get -(y1-y2)/-(x1-x2).

I hope you are aware of the rule that - signs can be cancelled out both a numerator and denominator.

So, we will be left with the formula, Slope = (y1-y2)/(x1-x2).

No worries Bunuel. The work that you and others are doing is anyways commendable since it is benefitting so many (including me)... Even though questions are taken from various sources, one would not go and check even if the source were known...

Hi guys, something I am missing.. why do we do a minus Xc (which is 12) as below for parallel lines???? is it always the case???

Slope AB=\frac{20-7}{5-30}=-052

For the line to be parallel to AB it will have the same slope, and will pass through a given point, C(12,10). We therefore have enough information to define the line by it's equation in point-slope form form:

y=-0.52(x-12)+10 --> y=-0.52x+16.24

The equation of a straight line that passes through a point \(P_1(x_1, y_1)\) with a slope \(m\) is:

\(y-y_1=m(x-x_1)\)

We calculated the slope \(m=-0.52\), and have the point \(C(12,10)\). substituting the values in the equation above we get: \(y-10=-0.52(x-12)\) or \(y=-0.52(x-12)+10\) as written.

I know the an absolute value of a slope gives us how steep the line would be. And the sign gives us whether it is a rise or a fall...

But if we have a question like: Line A has a slope -5 and Line B has a slope 4.... Which one of them has a greater slope? How do we handle this? Does this mean we consider the absolute values and then decide or answer.. (that is Line A)... or should we consider the signs too.. (i.e. Line B)...

Please advise!

If the question is which one has the greater slope, then the answer would be: Line B, as 4>-5. As you correctly noted line A will be steeper than B, but the slope of B is positive and that of A is negative. We are comparing m1 with m2 not |m1| with |m2|.
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Thank you so much for this chapter! It's very well written. One suggestion though - when we're actually doing GMAT question, and there is a fraction involved in the calculation, it is more often than not better to avoid converting it into decimal form until the absolute end of the question. Two reasons: 1. In Example #1 under Parellel lines section, it is not necessary to convert the slopes 14/29 and 20/-9 into decimal form. This is because the question requires us to figure out if the slopes are equal or not, and from the fraction form itself we can figure that out. 2. Often, you will be able to cancel out some parts of your fraction in a calculation that is to take place in the next step. For example, 9/2 is x. Find 2x. Answer: 9. (too easy example, but i hope u get the point.)
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My Practice GMAT Scores 29th Jan '11 -- GMATPrep#2 : 700 (Q47 V38) 23rd Jan '11 -- MGMAT Practice Test #3 : 670 (Q45 V36) 19th Jan '11 -- GMATPrep#1 v.1 : 710 (Q49 V37) 15th Jan '11 -- GMATPrep#1 : 720 (Q47 V42) 11th Jan '11 -- MGMAT Practice Test #2 : 740 (Q47 V44) 6th Jan '11 -- Kaplan#2 : 620 (Q40 V35) 28th Dec '10 -- PowerPrep#1 : 670 (Q47 V35) 30th Oct '10 -- MGMAT Practice Test #1 : 660 (Q45 V35) 12th Sept '10 -- Kaplan Free Test : 610 (Q39 V37) 6th Dec '09 -- PR CAT #1 : 650 (Q44 V37) 25th Oct '09 -- GMATPrep#1 : 620 (Q44 V34)

If you feel like you're under control, you're just not going fast enough. A goal without a plan is just a wish. You can go higher, you can go deeper, there are no boundaries above or beneath you.

Thanks So Much Man. A one point place for reviewing Coordinate Geometry. I have read most of these during EAMCET(Entrance Exam In A.P, India) time , now recollecting all, thanks to you. It would definitely take a long time to Google and learn all these, since most of the books dont cover so deep of a subject. Thanks Again and Keep on doing the great work.

Does anyone know how often the GMAT asks for anything beyond basic distance/slope? I have a hard time with these, especially rembering all the formulas, I have never seen a parabola question for example. Is it likely I would need to have this formula memorized?
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and 2ax+2by+2c = 0 and question is- are they parallel? I know both are same lines, but do we can them parallel?

In DS question the answer of- are they parallel should be NO? or Yes.

Basically you are asking whether the line is parallel to itself. It depends how we define the word "parallel". I don't think that there is a consensus about this issue nor that this concept is tested on GMAT. So don't worry about it.
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Could you please explain how we can solve the below question. If two lines are intersecting at point (10,27) and equation of one line is y=3x-3 What is the equation of another line.

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