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If $x \geq 0$ and $y \geq 0$, how many lattice points does the line $y = -2x + 18$ pass through? (A lattice point is a point with integer coordinates.)

gueesstt  May 4, 2018
 #1
avatar+91378 
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Since  x , y  ≥ 0.......we are only concerned with the lattice points intersected by the line in the first quadrant

 

So  we  have

 

y  = -2x + 18

The line has a y intercept  of  18.....so one lattice point  is  (0,18)

And letting  y  = 0, we can find the x intercept as

0 = -2x + 18

-18  = -2x

9  =  x ....so  a second lattice point is (9,0)

 

Now.....the slope  is  - 2.....which  means that y will change by -2  when x changes by 1

So.....the other lattice points intercepted by the line are

(1, 16)    (2, 14)   (3, 12)  (4, 10)  (5, 8)  (6, 6)  ( 7, 4)  (8,2)

 

So....10 lattice points

 

See the graph here :

 

https://www.desmos.com/calculator/climjgy2hb

 

 

cool cool cool

CPhill  May 4, 2018

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