In the rectangular coordinate system shown above, does the

[mdh3000]

In the rectangular coordinate system shown above, does the line k (not shown) intersect quadrant II?

1) Slope of k is -1/6

2) The y-intercept of k is -6

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The correct answer is A (statement 1 alone is sufficient).
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I'm have trouble understanding what assumptions are appropriate. Should I assume that line k goes on infinitely in each direction? If that's the case, then yes, I can see how something with a negative slope will eventually intersect quadrant II.

If the slope were positive, then I guess you couldn't say for sure if it intersected quadrant II, with more information (like the x-intercept).

Am I missing something??

mdh3000

[mdh3000]

Nobody?

mdh

[RonPurewal]

have patience, we'll get around to you. the forums are inundated right now with a random deluge of students just beginning their summer studies, but we're on it.

--

yes, lines are infinite, by definition. if you want a finite piece of a line, that's called a segment.

note that this is not how we use the word 'line' in the real world, but mathematical terms regularly use common words in different ways.
for instance, the word 'or', in mathematics, ALWAYS means 'inclusive or': i.e., A or B means A, or B, or both. this is not the case in the real world, where 'or' is sometimes inclusive (cream or sugar?) and sometimes exclusive (stop or i'll shoot!)

from your text it appears that you understand the solution to this problem, except for the issue of how to define a line. if this is not the case, and there are aspects of the problem you don't understand, feel free to post back and inquire.

[sara.leighton]

Can you explain how you know the line intersects the quadrant based on the slope only?

[daynekatz]

If the line had a slope of zero you would want to know the y intercept to determine which quadrants it passes through. However, if the line has a positive or negative slope by definition it will pass through atleast two quadrants, perhaps more: 2+4 (negative slope) or 1+3 (positive slope).

[RonPurewal]

Can you explain how you know the line intersects the quadrant based on the slope only?

any line with a negative slope goes up to the left, and down to the right, at a constant angle.
forever.

imagine that you have a line with a negative slope, then.
pick ANY starting point on this line.
if you go far enough to the left, starting at this point, the line WILL rise into the second quadrant. if the slope of the line is very gentle (i.e., almost flat) and the starting point is way, way deep below the x-axis, then the line might take a REALLY long time to get up above the x-axis, but it will get there.
for the same reason, but going to the right instead, the line WILL also eventually get into the fourth quadrant.

make a bunch of sketches if you don't see why this stuff has to be true.

remember, though, that LINES DON'T STOP. for the lines in your sketch with the gentlest and steepest slopes, you may well have to picture imaginary extensions of those lines, waaaaaaaayyyy off the sheet of paper on which you're actually drawing, in order to see how they're going to make it into quadrant 2 (or 4).

[jonathanc]

Hello,

Tough I understand Ron explication there is something unclear to me.

Is it wrong to assume that in the formula y=ax + b b can also be - ∞ ? In this case y = ax + b will not reach quadrant II whatever the slope is, will it ?

When Replied this question I did choose the answer C because I was thinking that the second sentence could help to dismiss this case especially because nowhere in the question was written than b should be an integer (actually they were even not any mention of b as a number which did reinforce my doubts.)

Is is wrong to assume that b can be ∞ or even in this case the line should reach quadrant II ?

I know this is borderline but I "often" (not that often but I am very upset when it s happening) have this kind of borderline thought when answering questions, it seems that what I was taught at school is not always the same as what is teached in the US and I am trying to have a list of all the differences I can meet. (e.g for the triangle what I learned in school was a≤b+c and not a<b+C and when a=b+c we have a specific triangle called flat triangle so first time I had a data sufficiency question requesting this inequality i did choose the wrong answer assuming the a=b+c was also possible fortunately this is not hard to remember)

Thank you,

John

[RonPurewal]

Hello,

Tough I understand Ron explication there is something unclear to me.

Is it wrong to assume that in the formula y=ax + b b can also be - ∞ ? In this case y = ax + b will not reach quadrant II whatever the slope is, will it ?

When Replied this question I did choose the answer C because I was thinking that the second sentence could help to dismiss this case especially because nowhere in the question was written than b should be an integer (actually they were even not any mention of b as a number which did reinforce my doubts.)

Is is wrong to assume that b can be ∞ or even in this case the line should reach quadrant II ?

editor:
infinity is not a number. you can NEVER let ANY quantity equal infinity, whether on this test or on any other algebra application.

extra credit:
if you've had experience in calculus, you'll know that even the actual definitions of "infinity" avoid the use of infinity altogether. for instance, "x approaches infinity" is defined formally in terms of x being larger than an arbitrary positive integer M.

in other words, the idea of a quantity called "infinity" is nonexistent in actual mathematics (except in the mathematics of transfinite cardinals, which is weird and not at all related to "normal" math).
the idea of infinity certainly drives a lot of math, but you can't use infinity as a quantity.
ever.

I know this is borderline but I "often" (not that often but I am very upset when it s happening) have this kind of borderline thought when answering questions, it seems that what I was taught at school is not always the same as what is teached in the US and I am trying to have a list of all the differences I can meet. (e.g for the triangle what I learned in school was a≤b+c and not a<b+C and when a=b+c we have a specific triangle called flat triangle so first time I had a data sufficiency question requesting this inequality i did choose the wrong answer assuming the a=b+c was also possible fortunately this is not hard to remember)

Thank you,

John

editor:
if you're talking about GEOMETRY, then it is a universal mathematical fact that "flat triangles" are excluded from consideration as triangles. i.e., "flat triangles" are NOT triangles; they're just line segments that are on top of each other.

the "less than or equal to" inequality you're describing applies to some other areas, such as the calculus of vectors, but it does not apply to geometry.

[jyothi h]

Hi Ron ,

I had a doubt regarding statement 2 . when we say that the y intercept of a line is so and so , it would mean that there is only one point on the x axis that the line intersects . If we go by this understanding ,then isnt option 2 also sufficient ? I am a little unclear on option 2 . If the line falls on the x axis ( rather is x-axis itself) , then are we still allowed to pick a point on x axis and say that the y intercept on the line is this value . I thought , option B translates to - "there is only one point -6 , on x-axis that this line intercepts"
Appreciate if any of the instructors can help clarify my doubt.

Thanks,
Jyothi

[RonPurewal]

The y-intercept is where a line crosses the y-axis.

[jyothi h]

The y-intercept is where a line crosses the y-axis.

Oops , I meant to write x-intercept .Sorry about that . Dont know what I was thinking when I wrote this . Below is the corrected version of my post .

---- I had a doubt regarding statement 2 . when we say that the X intercept of a line is so and so , it would mean that there is only one point on the x axis that the line intersects . If we go by this understanding ,then isnt option 2 also sufficient ? I am a little unclear on option 2 . If the line falls on the x axis ( i.e line is x-axis itself) , then are we still allowed to pick ONE point on x axis and say that the X intercept on the line is this value . I thought , option B translates to - "there is only one point -6 , on x-axis that this line intercepts"

Thanks,
Jyothi

[RonPurewal]

Statement 2 talks about the y-intercept. It says nothing about the x-intercept.

[jyothi h]

Statement 2 talks about the y-intercept. It says nothing about the x-intercept.

I am EXTREMELY sorry that I posted back with another mistake. I tend to mix up x and y intercepts often. Need to work on it .
Thanks a lot for correcting me and apologize again for having taken your time on this.

-Jyothi

[RonPurewal]

Nothing to apologize for -- just note the mistake, and work on paying more attention to that particular thing when you see it next time. Sometimes people don't even realize which things need more attention until they post errors like this.

I have the same problem myself -- in fact, I'm dyslexic, so I probably switch letters a lot more often than you do. But I know it's a problem, so I just triple-check x vs. y every single time I'm working on a problem that involves coordinates.

[jyothi h]

Nothing to apologize for -- just note the mistake, and work on paying more attention to that particular thing when you see it next time. Sometimes people don't even realize which things need more attention until they post errors like this.

I have the same problem myself -- in fact, I'm dyslexic, so I probably switch letters a lot more often than you do. But I know it's a problem, so I just triple-check x vs. y every single time I'm working on a problem that involves coordinates.

Sure , I'll note your advice and work on it.
Thanks again !