SOLUTION: I need the work shown for this: Find two points such that their coordinates are integers (positive AND negative numbers) and the distance between them is 10.

You can put this solution on YOUR website!
the length of a line in coordinate geometry is:

if L = 10, this means that:

if you square both sides of this equation you get:

let A = (x2-x1)^2
let B = (y2-y1)^2
equation becomes:
100 = A^2 + B^2
since the coordinates have to be integers, this restricts the choices.
if they didn't have to be integers, you could just have picked any number less than 100 but greater than 0 and the other number would be 100 minus that.
---
since they have to be integers, then only certain combinations would do:
ones that i know of right off the bat are:
36 and 64
square root of 36 is 6
square root of 64 is 8
this would mean that:
x2-x1 = 6
y2-y1 = 8
or vice versa.
since you only need one, this will do.
---
now that you have this relationship, you can pick any x value or y value at random and calculate so that the length of each of this is as stated.
---
take x2-x1 = 6
let x2 = 3
that means that x1 have to be = to -3
---
take y2 = 6
that means that y1 must = -2
---
3 - (-3) = 6 so the x values are good.
6 - (-2) = 8 so the y values are good.
---
your coordinate points are:
(x2,y2) = (3,6)
(x1,y1) = (-3,-2)
---
to prove these values are good, substitute in the equation for the length of a line segment.
that equation is:

(y2-y1) = 8
(x2-x1) = 6
equation becomes:

which becomes:

which becomes:

which becomes: