Question

A line passes through `(2 ,8 )` and `( 3, 5 )`. A second line passes through `( 4, 4 )`. What is one other point that the second line may pass through if it is parallel to the first line?

Answer 1

One possible point would be `(6,-2)`

Explanation:

A line through `(2,8)` and `(3,5)` has a slope of
`color(white)("XXX")(Deltay)/(Deltax)=(5-8)/(3-2)=(-3)/1`

Any line parallel to this will have the same slope.

Any point on a line through `(4,4)` with the form
`color(white)("XXX")(4+k * Deltax,color(white)("XX")4+k * Deltay)`
with a non-zero constant `k` and `Deltax=1, Deltay=-3`
will have a slope of
`color(white)("XXX")(k * Deltay)/(k * Deltax)=(k * (-3))/(k *1)=(-3)/1`
and will therefore be parallel to the line through `(2,8)` and `(3,5)`

Arbitrarily picking `k=2`, one such point would be
`color(white)("XXX")(4+2 * Deltax,color(white)("XX")4+2 * Deltay)`
`color(white)("XXXXXX") = (4+2 * 1,color(white)("X")4+2 * (-3))`
`color(white)("XXXXXX")=(6, -2)`