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Line Q has the equation 5y – 3x = 45. If Line S is perpendicular to Q

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Line Q has the equation 5y – 3x = 45. If Line S is perpendicular to Q [#permalink]

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Line Q has the equation 5y – 3x = 45. If Line S is perpendicular to Q, has an integer for its y-intercept, and intersects Q in the second quadrant, then how many possible Line S’s exist? (Note: Intersections on one of the axes do not count.)
(A) 25
(B) 33
(C) 36
(D) 41
(E) 58


For a bank of challenging coordinate geometry problems, as well as the OE to this one, see:
http://magoosh.com/gmat/2014/challengin ... questions/

Mike :-)
[Reveal] Spoiler: OA

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Re: Line Q has the equation 5y – 3x = 45. If Line S is perpendicular to Q [#permalink]

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New post 30 Aug 2014, 00:02
5y - 3x = 45 --> 5y = 3x + 45
at x= 0, y = 9
at y=0, x = -15
Now slope = 3/5
perpendicular 's slope will be -5/3
at ( -15,0) perp line eqn = y-0 = -5/3(x-(-15))
thus y = -5/3 x - 25
thus at x=0, y = - 25
24 + 1 + 8 = 33

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Re: Line Q has the equation 5y – 3x = 45. If Line S is perpendicular to Q [#permalink]

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New post 30 Aug 2014, 21:14
How do we get 24 + 1 + 8 = 33
Please explain?

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Re: Line Q has the equation 5y – 3x = 45. If Line S is perpendicular to Q [#permalink]

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RonnieRon wrote:
How do we get 24 + 1 + 8 = 33
Please explain?

Dear RonnieRon,
Please see a full explanation here:
http://magoosh.com/gmat/2014/challengin ... questions/
It's #7 on that page.
Mike :-)
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Re: Line Q has the equation 5y – 3x = 45. If Line S is perpendicular to Q [#permalink]

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New post 30 Aug 2014, 23:57
Finally get the concept but a little too difficult for myself

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Line Q has the equation 5y – 3x = 45. If Line S is perpendicular to Q [#permalink]

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New post 31 Aug 2014, 17:52
Hi Mike
thanks - great explanation...+1

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Line Q has the equation 5y – 3x = 45. If Line S is perpendicular to Q [#permalink]

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Line Q has the equation 5y – 3x = 45. If Line S is perpendicular to Q, has an integer for its y-intercept, and intersects Q in the second quadrant, then how many possible Line S’s exist? (Note: Intersections on one of the axes do not count.)
(A) 25
(B) 33
(C) 36
(D) 41
(E) 58


The equation of the given line Q is \(5y-3x=45\) or \(y = \frac{3}{5}x +9\)

Therefore, slope of the line Q = \(\frac{3}{5}\)

Slope of line S, which is perpendicular to line Q = \(\frac{-5}{3}\)

X-intercept of line Q = \(-15\)

Highest possible value of Y-intercept of the line S that is perpendicular to line Q and that intersects in the second quadrant
= Y-intercept of the line Q =\(9\)

Lowest possible value of Y-intercept of the line that is perpendicular to line Q and that intersects in the second quadrant
= - (slope of line Q) *(X-intercept of line Q or X-intercept of line S that intersects Q at X-axis=-15) = \(-25\)

The number of possible perpendicular line S that intersects line Q in the second quadrant and that its Y-intercept is a integer quantity (excluding intersection at axes)= Integer values between 9 and -25= 9 – (-25)-1 =9+25-1 =\(33\)

Answer: (B)
Attachments

Possible-Perpendicular-Lines.png
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Last edited by arunspanda on 10 Apr 2016, 04:51, edited 1 time in total.

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Re: Line Q has the equation 5y – 3x = 45. If Line S is perpendicular to Q [#permalink]

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New post 12 Jul 2016, 01:27
Hello Mike,

Well this was a difficult question for me.

I understood the solution but i have a silly doubt-intersection at the axes means the perpendicular line S that passes through the origin should be excluded??

:roll:

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Re: Line Q has the equation 5y – 3x = 45. If Line S is perpendicular to Q [#permalink]

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Ashishsteag wrote:
Hello Mike,

Well this was a difficult question for me.

I understood the solution but i have a silly doubt-intersection at the axes means the perpendicular line S that passes through the origin should be excluded??

:roll:

Dear Ashishsteag,

I'm happy to respond. :-) Yes, this is an extremely difficult question.

Your question is not silly. Let's look at the prompt:
Line Q has the equation 5y – 3x = 45. If Line S is perpendicular to Q, has an integer for its y-intercept, and intersects Q in the second quadrant, then how many possible Line S’s exist? (Note: Intersections on one of the axes do not count.)
In this problem, the intersection of concern, the only intersection being discussed, is the intersection of Line Q and Line S. That is the only intersection that matters in the problem. If these two lines intersect on the axis---for example, at the point (0, 9)---then we wouldn't count that as an intersection in QII, precisely because points on either axis are not points in QII. Line S will have its own intersections with the x- and y-axes, but those are irrelevant to the problem. It doesn't matter where Line S intersects the axes, even at the origin. The only thing that matters is the intersection point of Line Q and line S.

Does all this make sense?
Mike :-)
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Re: Line Q has the equation 5y – 3x = 45. If Line S is perpendicular to Q [#permalink]

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New post 12 Jul 2016, 18:30
mikemcgarry wrote:
Ashishsteag wrote:
Hello Mike,

Well this was a difficult question for me.

I understood the solution but i have a silly doubt-intersection at the axes means the perpendicular line S that passes through the origin should be excluded??

:roll:

Dear Ashishsteag,

I'm happy to respond. :-) Yes, this is an extremely difficult question.

Your question is not silly. Let's look at the prompt:
Line Q has the equation 5y – 3x = 45. If Line S is perpendicular to Q, has an integer for its y-intercept, and intersects Q in the second quadrant, then how many possible Line S’s exist? (Note: Intersections on one of the axes do not count.)
In this problem, the intersection of concern, the only intersection being discussed, is the intersection of Line Q and Line S. That is the only intersection that matters in the problem. If these two lines intersect on the axis---for example, at the point (0, 9)---then we wouldn't count that as an intersection in QII, precisely because points on either axis are not points in QII. Line S will have its own intersections with the x- and y-axes, but those are irrelevant to the problem. It doesn't matter where Line S intersects the axes, even at the origin. The only thing that matters is the intersection point of Line Q and line S.

Does all this make sense?
Mike :-)



Actually,I got confused with this part of the question written at the very end:"(Note: Intersections on one of the axes do not count.)",and then the question also speaks about y-intercept in the part:"If Line S is perpendicular to Q, has an integer for its y-intercept",so I thought that since the perpendicular line S does not have any sort of y-intercept at the origin (0,0),so we need to exclude that part?
Even one of the solutions mentioned above with the required figure drawn is subtracting 1 at the very end of the solution.I understand your point that only intersections between line S and line Q need to be considered and they need not be counted on either of the axes.Thanx a lot for your help. :)

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Re: Line Q has the equation 5y – 3x = 45. If Line S is perpendicular to Q [#permalink]

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New post 05 Nov 2016, 06:06
my question is - we calculated the intersections in the y axis between 9 and -25 to get the answer; why did we not calculate the intersections in the x axis between -15 and 5 instead? Please bear with me if this is a stupid question. Thank you.

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TheLordCommander wrote:
my question is - we calculated the intersections in the y axis between 9 and -25 to get the answer; why did we not calculate the intersections in the x axis between -15 and 5 instead? Please bear with me if this is a stupid question. Thank you.

Dear TheLordCommander,

I'm happy to respond. :-) This is a hard question, so questions about it are not "stupid." :-)

Part of the requirement of the question is that the y-intercept has to be an integer. If we mark off the boundaries on the y-intercept, then we simply can count integers along the y-axis.

You see, if the x-intercept is an integer, that doesn't guarantee that the y-intercept is an integer (unless the slope is +1 or -1). Certainly for any non-integer slope, the general rule is that for most x-intercepts that are integers, the y-intercept is not an integer, and vice versa. If we start looking at points on the x-axis, we know they have to be -15 and 5, but within that range, we have no idea what spacings of the values of the x-intercept would result in integer values on the y-intercept.

Therefore, it's much easier simply to stick to the y-intercept and count integers.

Does all this make sense?
Mike :-)
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Re: Line Q has the equation 5y – 3x = 45. If Line S is perpendicular to Q [#permalink]

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New post 05 Nov 2016, 11:10
mikemcgarry wrote:
TheLordCommander wrote:
my question is - we calculated the intersections in the y axis between 9 and -25 to get the answer; why did we not calculate the intersections in the x axis between -15 and 5 instead? Please bear with me if this is a stupid question. Thank you.

Dear TheLordCommander,

I'm happy to respond. :-) This is a hard question, so questions about it are not "stupid." :-)

Part of the requirement of the question is that the y-intercept has to be an integer. If we mark off the boundaries on the y-intercept, then we simply can count integers along the y-axis.

You see, if the x-intercept is an integer, that doesn't guarantee that the y-intercept is an integer (unless the slope is +1 or -1). Certainly for any non-integer slope, the general rule is that for most x-intercepts that are integers, the y-intercept is not an integer, and vice versa. If we start looking at points on the x-axis, we know they have to be -15 and 5, but within that range, we have no idea what spacings of the values of the x-intercept would result in integer values on the y-intercept.

Therefore, it's much easier simply to stick to the y-intercept and count integers.

Does all this make sense?
Mike :-)


Makes a lot of sense Mike. Thanks :)

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Re: Line Q has the equation 5y – 3x = 45. If Line S is perpendicular to Q [#permalink]

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