Question

A line segment is bisected by a line with the equation `-3 y + 7 x = 1`. If one end of the line segment is at `(9 ,2 )`, where is the other end?

Answer 1

The point `( -262/58,452/58)`

Explanation:

Write the given equation of the bisector line in slope-intercept:

`y = 7/3x - 1/3` [1]

The slope of the line that goes through point `(9,2)` is the negative reciprocal of the slope of the bisector line, `-3/7`. Use the point-slope form of the equation of a line to write the equation for the line segment:

`y - 2 = -3/7(x - 9)`

`y - 2 = -3/7x + 27/7`

`y = -3/7x + 27/7 + 2`

`y = -3/7x + 41/7` [2]

To find the x coordinate of the point of intersection subtract equation [2] from equation [1]

`y - y= 7/3x + 3/7x- 1/3 - 41/7`

`0 = 58/21x - 130/21`

`x = 130/58`

The change in x from 9 to `214/58` is:

`Deltax = 130/58 - 9`

`Deltax = -392/58`

The x coordinate for other end will have twice that change:

`x = 9 - 784/58`

`x = -262/58`

To find the y coordinate of the other end, substitute the x coordinate into equation [2]

`y = -3/7(-262/58) + 41/7`

`y = 452/58`