A line segment is bisected by a line with the equation `3 y - 7 x = 2`. If one end of the line segment is at `(7 ,8 )`, where is the other end?

Rewrite the given line in slope-intercept form so that we can obtain the slope, m, of the line.

`y = 7/3x + 2/3` `color(white)_[1]`

We observe that `m = 7/3`

The slope, n, of the bisected line is the negative reciprocal of m:

`n = -1/m`

`n = -3/7`

Use this slope and the given point to solve for `b` in the slope-intercept form, `y = nx + b`:

`8 = -3/7(7) + b`

`b = 11`

The equation of the bisected line is:

`y = -3/7x + 11` `color(white)_[2]`

Subtract equation [2] from equation [1]:

`y- y = 7/3x + 3/7x + 2/3 - 11`

`0 = 58/21x - 31/3`

`58/21x = 31/3`

The x coordinate of the point two lines intersect is: `x = 217/58`

The change in x from the point `(7,8)` to the point of intersection is:

`Deltax = 217/58 - 7`

`Deltax = -189/58`

To go to the other end of the line we must move twice that far in same direction:

`2Deltax = -378/58`

The add 7 to find the x coordinate of the other end of the line segment:

`7 + 2Deltax = 7 -378/58 = 28/58 = 14/29`

To find the y coordinate of the other end of the line segment substitute 14/28 for x in equation [2]:

`y = -3/7(14/29) + 11`

`y = 313/29`