A line segment is bisected by a line with the equation `- y + 4 x = 3`. If one end of the line segment is at `( 2 , 6 )`, where is the other end?

Consider the vertical line `color(magenta)(x=2)`
Clearly `color(magenta)x=2` goes through the point `color(red)(""(2,6))`
and it intersects `color(blue)(-y+4x=3)` at `color(blue)(""(2,5))`

`color(red)(""(2,6))` is vertically `color(brown)(1)` unit above `color(blue)(""(2,5))`, the intersection point with `color(blue)(-y+4x=3)`

`color(green)(""(2,4))` is vertically `color(brown)1` unit below `color(blue)(""(2,5))`, the intersection point with `color(blue)(-y+4x=3)`

Therefore `color(blue)(-y+4x=3)` bisects the line segment between `color(green)(""(2,4))` and `color(red)(""(2,6))`

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Furthermore (as seen in the image below and considering similar triangles)
any point on a line parallel to `color(blue)(-y+4x=3)` through `color(green)(""(2,4))` will provide, together with `color(red)(""(2,6))` a line segment bisected by `color(blue)(-y+4x=3)`

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

`color(blue)(-y+4x=3)` can be written in slope-intercept form as
`color(white)("XXX")color(blue)(y=4x-3)` with a y-intercept at `color(blue)(""(-3))`

If the line through `color(green)(""(2,4))` parallel to `color(blue)(-y+4x=3)` is `color(brown)(1)` unit vertically below `color(blue)(-y+4x=3)`
it will have a y-intercept at `color(green)(""(-4))`
and therefore a slope-intercept form of
`color(white)("XXX")color(green)(y=4x-4)`