A line segment is bisected by a line with the equation `2 y + 3 x = 3`. If one end of the line segment is at `( 1 , 8 )`, where is the other end?

The equation of any line segment that has the line,

`3x + 2y = 3" [1]"`

, as its perpendicular bisector will have the standard form:

`2x - 3y = C`

To find the value of C for the specified line segment, substitute the point `(1, 8)` into the equation:

`2(1) - 3(8) = C`

And then solve for C:

`C = -22`

The equation of the bisected line segment is:

`2x - 3y = -22" [2]"`

We shall use equations [1] and [2] to find the x coordinate of the point of intersection, `x_i`.

Multiply equation [1] by 3 and equation [2] by 2:

`9x_i + 6y_i = 9" [3]"`
`4x_i - 6y_i = -44" [4]"`

Add equation [3] to equation [4]:

`13x_i = -35`

`x_i = -35/13`

The change in x (`Deltax)`from the starting point to the point of intersection is as follows:

`Deltax = (-35/13 - 1)`

`Deltax = -48/13`

The x coordinate of the other end, `x_e` can be computed as follows:

`x_e = 2Deltax + x_s`

where `x_s` is the starting x coordinate, `x_s = 1`:

`x_e = 2(-48/13) + 1`

`x_e = -83/13`

To find the corresponding y coordinate, `y_e` substitute `x_e` into equation [2]:

`2(-83/13) - 3y_e = -22`

`y_e = 40/13`

To other end is at the point `(-83/13,40/13)`

Here is a graph with the two lines and the start and end points plotted:

![Desmos.com]()