For x between -100 and 100, inclusive, in the equation
y= - 4/3 x + 70/3
do you consider a lattice point to be a point {x,y} where x,y are integers? Lattices can be however you define them.
If so
how often will an x value produce an integer y value given the denominator of both terms in y(x)?
How many integer values of x are there for {-100 <= x <= 100}
So roughly how many lattice points will there be?
Now look at the edges of your x domain and make sure you don't have to subtract 1.
-4(-98)+70 = 462 which is divisible by 3, so they start at -98
-4(100)+70 = -330 which is divisible by 3 so they stop at 100
You can figure out how many there are from all this.
I'm thinking it's 32 + 33, but I get so confused on this kind of thing because, from 0-20, there are 7 x's that work, but if you just divide 20 by 3 you only get 6 with some remainder. If you divide 100 by 3 you have a remainder of 1, I don't think that's enough to get another number though?
Oh, I think I got a way - line up all the numbers (positives first) that work for x... you get 1, 4, 7, 10.... 100. Add 2 to everything and you get multiples of 3, then divide everything by three to get 1, 2, 3... etc. and you have the same AMOUNT of numbers. You end up with 34 for numbers up to 100 (102/3=34).
Same thing starting from -98 and ending at -3.... add -1 to each number to get -99, -96,..., -3 and divide by -3 to find there are 99/3= 33 numbers. I won't count 0 because 0+70 isn't divisible by 3.
So all in all we have 34 + 33?
you're killin me. Who cares what the y values are. You just need to find the number of x's so that -4x+70 is divisible by 3.
Once you find your starting point that's every 3rd x. We know the starting point. It's -98 given your range of -100,100.
so -98, -95, -93 ... 94, 97, 100
are your x points. Now just count them. Like I said before 100 = -98 + 3*66, but -98 is included as well so you have 67 total points.
Hello, orange!
$\displaystyle \text{On }[\text{-}100, 100],\text{ how many lattice points are on the line: }y \:=\:-\tfrac{4}{3}x + \tfrac{70}{3}$
We have: .$\displaystyle y \:=\:\text{-}x - \frac{x}{3} + 23 + \frac{1}{3} \quad\Rightarrow\quad y \:=\:\text{-}x + 23 + \frac{1-x}{3}$
Since $\displaystyle y$ is an integer, then $\displaystyle 1-x$ must be a multiple of 3.
. . That is: .$\displaystyle 1 - x \:=\:3a\:\text{ for some integer }a$
We have: .$\displaystyle x \:=\:1-3a$
Since $\displaystyle x \ge \text{-}100\!:\;\;1-3a \,\ge\,\text{-}100 \quad\Rightarrow\quad \text{-}3a\,\ge\,\text{-}101 \quad\Rightarrow\quad a\,\le\,33$
Since $\displaystyle x \le 100\!:\;\;1-3a\,\le\,100 \quad\Rightarrow\quad \text{-}3a \,\le\,99 \quad\Rightarrow\quad a \,\ge\,\text{-}33$
. . Thus: .$\displaystyle a \in [\text{-}33,\,33]$
Then $\displaystyle a$ (and hence, $\displaystyle x$) can take on 67 values.
Therefore, there are $\displaystyle 67$ lattice points on the line segment.