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204. How many integer points lie between points A and В on t

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204. How many integer points lie between points A and В on t [#permalink] New post 18 Feb 2011, 09:25
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204. How many integer points lie between points A and В on the line segment AB, if A is (5, 7) and В is (10, –3)?
(A) 4
(B) 5
(C) 6
(D) 10
(E) 15
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Re: 204. How many integer points lie between points A and В on t [#permalink] New post 18 Feb 2011, 09:55
equation of line passing through two points (x1,y1) and (x2,y2)

y-y_1=\frac{y_2-y_1}{x_2-x_1}(x-x_1)

(x_1,y_1) = (5,7)
(x_2,y_2) = (10,-3)

y-7=\frac{-3-7}{10-5}(x-5)

y-7=-2(x-5)

y+2x=17

Plug in values for x from 6 to 9 and see whether y will be an integer.

x=6; y=5
x=7; y=3
x=8; y=1
x=9; y=-1

Total 4 points that are integers.

Ans: "A"
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Re: 204. How many integer points lie between points A and В on t [#permalink] New post 18 Feb 2011, 10:07
Expert's post
banksy wrote:
204. How many integer points lie between points A and В on the line segment AB, if A is (5, 7) and В is (10, –3)?
(A) 4
(B) 5
(C) 6
(D) 10
(E) 15


You can notice that while we moved 5 units to the right (from x=5 to x=10) we moved 10 units down (from y=7 to y=-3) thus every unit to the right = exactly 2 units down. So each of 4 integer coordinates of x between 5 and 10 (6, 7, 8 and 9) will have an integer pair of y (5, 3, 1, and -1).

To demonstrate:
x=5, y=7;
x=5+1=6 --> y=7-2=5;
x=6+1=7 --> y=5-2=3;
x=7+1=8 --> y=3-2=1;
x=8+1=9 --> y=1-2=-1;
x=9+1=10 --> y=-1-2=-3.

So there are 4 points with integer co-ordinates: (6, 5), (7, 3), (8, 1), and (9, -1).

Answer: A.
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Re: 204. How many integer points lie between points A and В on t   [#permalink] 18 Feb 2011, 10:07
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204. How many integer points lie between points A and В on t

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