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Line Q has the equation 5y – 3x = 45. If Line S is perpendicular to Q [#permalink]
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29 Aug 2014, 16:58
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Line Q has the equation 5y – 3x = 45. If Line S is perpendicular to Q, has an integer for its yintercept, 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) 58For a bank of challenging coordinate geometry problems, as well as the OE to this one, see: http://magoosh.com/gmat/2014/challengin ... questions/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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30 Aug 2014, 01:02
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 = y0 = 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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30 Aug 2014, 22:14
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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30 Aug 2014, 22:56
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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31 Aug 2014, 00:57
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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31 Aug 2014, 18:52
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 yintercept, 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 \(5y3x=45\) 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 = \(\frac{5}{3}\) Xintercept of line Q = \(15\) Highest possible value of Yintercept of the line S that is perpendicular to line Q and that intersects in the second quadrant = Yintercept of the line Q =\(9\) Lowest possible value of Yintercept of the line that is perpendicular to line Q and that intersects in the second quadrant =  (slope of line Q) *(Xintercept of line Q or Xintercept of line S that intersects Q at Xaxis=15) = \(25\) The number of possible perpendicular line S that intersects line Q in the second quadrant and that its Yintercept is a integer quantity (excluding intersection at axes)= Integer values between 9 and 25= 9 – (25)1 =9+251 =\(33\) Answer: (B)
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Originally posted by arunspanda on 01 Sep 2014, 03:32.
Last edited by arunspanda on 10 Apr 2016, 05:51, 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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12 Jul 2016, 02:27
Hello Mike, Well this was a difficult question for me. I understood the solution but i have a silly doubtintersection at the axes means the perpendicular line S that passes through the origin should be excluded??



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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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12 Jul 2016, 10:34
Ashishsteag wrote: Hello Mike, Well this was a difficult question for me. I understood the solution but i have a silly doubtintersection at the axes means the perpendicular line S that passes through the origin should be excluded?? 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 yintercept, 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 axisfor 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 yaxes, 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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12 Jul 2016, 19:30
mikemcgarry wrote: Ashishsteag wrote: Hello Mike, Well this was a difficult question for me. I understood the solution but i have a silly doubtintersection at the axes means the perpendicular line S that passes through the origin should be excluded?? 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 yintercept, 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 axisfor 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 yaxes, 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 yintercept in the part:"If Line S is perpendicular to Q, has an integer for its yintercept",so I thought that since the perpendicular line S does not have any sort of yintercept 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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05 Nov 2016, 07:06
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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Re: Line Q has the equation 5y – 3x = 45. If Line S is perpendicular to Q [#permalink]
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05 Nov 2016, 11:46
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 yintercept has to be an integer. If we mark off the boundaries on the yintercept, then we simply can count integers along the yaxis. You see, if the xintercept is an integer, that doesn't guarantee that the yintercept is an integer (unless the slope is +1 or 1). Certainly for any noninteger slope, the general rule is that for most xintercepts that are integers, the yintercept is not an integer, and vice versa. If we start looking at points on the xaxis, we know they have to be 15 and 5, but within that range, we have no idea what spacings of the values of the xintercept would result in integer values on the yintercept. Therefore, it's much easier simply to stick to the yintercept 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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05 Nov 2016, 12:10
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 yintercept has to be an integer. If we mark off the boundaries on the yintercept, then we simply can count integers along the yaxis. You see, if the xintercept is an integer, that doesn't guarantee that the yintercept is an integer (unless the slope is +1 or 1). Certainly for any noninteger slope, the general rule is that for most xintercepts that are integers, the yintercept is not an integer, and vice versa. If we start looking at points on the xaxis, we know they have to be 15 and 5, but within that range, we have no idea what spacings of the values of the xintercept would result in integer values on the yintercept. Therefore, it's much easier simply to stick to the yintercept and count integers. Does all this make sense? Mike Makes a lot of sense Mike. Thanks



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