A line segment is bisected by a line with the equation `4 y + x = 8`. If one end of the line segment is at `(5 ,6 )`, where is the other end?

A vertical line through `color(green)(""(5,6))` will intersect `color(purple)(4y+x=8)`
at `color(purple)(""(5,3/4))`

`color(purple)(""(5,3/4))` is `5 1/4` units below `color(green)(""(5,6)`

If `color(purple)(""(5,3/4))` is the bisect point of the vertical line,
the second end point must be `5 1/4` units lower than `color(purple)(""(5,3/4))`
That is the second end point must be at `color(red)(""(5,-5 1/2))`

The given line `color(purple)(4y+x=8)` bisects the vertical line segment between `color(green)(""(5,6))` and `color(red)(""(5,-5 1/2))`

Furthermore (based on similar triangles) any point on a line parallel to the bisector line `color(purple)(4y+x=8)` will also be bisected by `color(purple)(4y+x=8)`

`color(purple)(4y+x=8)` has a slope of `(-1/4)`
So all lines parallel to `color(purple)(4y+x=8)` will have a slope of `(-1/4)`

The equation for a line passing through `color(red)(""(5,-5 1/2))` with a slope of `(-1/4)` can be expressed in slope-point form as
`color(white)("XXX")color(red)(y- (-5 1/2) = -1/4(x-5))`
or after simplifying, as
`color(white)("XXX")color(red)(4y+x=-17)`