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In-Depth Coordinate Geometry - Expert Help needed!

Problem Solving — algebra and arithmetic (GMAT Focus Edition)
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In-Depth Coordinate Geometry - Expert Help needed!

by Tega1984 » Wed May 18, 2011 11:52 am
I found an article about coordinate geometry on GMATHACKS. In this article the author listed some interesting questions about this concept,the questions which I think anyone who's serious about GMAT and who wants to understand the concepts thoroughly should be able to answer. He left us to figure out the answers on our own. So I thought a bit about those questions and came up with some answers (I was not able to answer all of those) and now I need some expert help, in order to find out whether my answers are correct and if I'm truly on the way of mastering this concept. So here are these questions in blue and my answers in red. Hope you'll find this topic interesting.

What does a line with a positive slope less than one look like? I did not notice any particular pattern of this type of line,it seems like an ordinary line with a positive slope and that's it, any ideas?
What's the slope of a very steep line? - Not really sure about that one either, slope of infinity?
What's the slope of a line with negative x- and y-intercepts? Negative Slope.
How about a negative x-intercept and a positive y-intercept? Positive Slope
How do you determine whether a certain point is on the line? If we have the slope and y-intercept of that line, we can plug in the coordinates of that point into the slope intercept form and if these coordinates satisfy the equation, than the point lies on that line.
What does it mean if "m" is a positive fraction?-It means that the slope is positive
What if "m" is negative?-The slope is negative
What happens if you increase "b"? The x-intercept will decrease
What happens if we Decrease "b"? The x intercept will increase

I looking forward to the experts answers.
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by djiddish98 » Wed May 18, 2011 11:58 am
My thoughts (I am not an expert, but I wanted to take a crack at it)


What does a line with a positive slope less than one look like? It's going to be increasing, but much flatter than > 1. We're moving 1 "y-space" for every z "x-spaces", if the slope is 1/z.
What's the slope of a very steep line? As the line gets steeper, we eventually end up with an undefined slope. We have a huge change in Y with no change in X. A straight vertical line has an undefined slope, since the change in x = 0, and we'd be dividing by 0.
What's the slope of a line with negative x- and y-intercepts?

What happens if you increase "b"? The Y-intercept will increase
What happens if we Decrease "b"? The Y intercept will decrease

I assume for the last two that the equation is setup as y = mx + b. B is then the Y-intercept, since when x = 0, y = b.

The x-intercept can be derived by calculating x = (y-b)/m. If Y = 0, then x = -b/m. So the x intercept is a function of b AND m.

for example, if b was 3 and m was -1/3, then the x intercept would increase if we increased b to 4.

-3/(-1/3) = 9 /// -4/(-1/3) = 12

Please feel free to correct me if I'm wrong.
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by Tega1984 » Thu May 19, 2011 12:06 am
Thanks for the reply.Yeah I agree with you on everything actually. I just wanted to add that when you increase/decrease b, the x-intercept also changes it's value. Since b stands for Y-intercept,it's obvious that Y-intercept increases when b increases and Y-intercept decreases when b decreases. However,there's more than that!You can check for yourself and you'll find out that as b increases at the same time the X-intercept will decrease, and as b decreases the X-intercept will increase.

P.S. I thought that the X-intercept can be derived by setting y=0, I didn't know about x=(y-b)/m.
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by djiddish98 » Thu May 19, 2011 3:53 am
You are correct that x the intercept does change - we're just not sure whether it increases or decreases.

You are also right that the x-intercept is when y = 0. It's just that we still have 2 variables when y = 0; b and m.

if y = mx + b, we can rearrange to isolate x -> x = (y-b)/m. If we set y = 0, then x = -b/m.

When x is zero, it's easier, because it takes out the m variable and we're left with y = b.
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by Tani » Sat May 21, 2011 7:31 am
The easy way to identify >1 versus <1 slopes is to picture a slope of 1 for a line that cuts through the origin. That line will form an angle of 45 degrees with each axis. Looking at any other line going through the origin, if the line in quadrant one is below our original line, the slope is less than one. If the line in quadrant one is above our original line, the slope is greater than one.
Tani Wolff
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