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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.)

May 4, 2018

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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

May 4, 2018