Question

A line segment is bisected by a line with the equation `4 y - 3 x = 2`. If one end of the line segment is at `( 2 , 5 )`, where is the other end?

Answer 1

The other end will be any point on the line `4y-3x=-10`

Explanation:

Consider the vertical line `x=2` which passes through `(2,5)`

`x=2` will intersect `4y-3x=2` at `(2,2)`

`(2,5)` is `3` units vertically above `(2,2)`;
that is `(2,5)` is vertically `3` units above `4y-3x=2`

Any point `3` units vertically below `4y-3x=2` will provide a second point which together with `(2,5)` form a line segment bisected by `4y-3x=2`

Re-writing `4y-3x=2` in slope-intercept form: `y=3/4x+2/4`

The y-intercept for a line `3` units below `3=3/4x+2/4` will be at `2/4-3=-10/4`

Therefore this second line will have a slope-intercept equation of
`color(white)("XXX")y=3/4x-10/4`
or in a form similar to the initial equation:
`color(white)("XXX")4y-3x=-10`

The image below may help: