Question

A line segment is bisected by a line with the equation `4 y + 3 x = 4`. If one end of the line segment is at `( 8 , 9 )`, where is the other end?

Answer 1

The other end is the point `(-168/25, -223/25)`

Explanation:

Write the equation of the bisector is slope-intercept form:

`y = -3/4x + 1color(white)_[1]`

The slope of the bisected line is the negative reciprocal, `4/3`. Use the point-slope form of the equation of a line to force the line through point `(8,9)`

`y - 9 = 4/3(x - 8)`

`y - 9 = 4/3x - 32/3`

`y = 4/3x - 32/3 + 9`

`y = 4/3x - 5/3color(white)_[2]`

Subtract equation [1] from equation [2]:

`0 = (4/3 + 3/4)x - 8/3`

Solve for x:

`8/3 = 25/12x`

This is the x coordinate of the point of intersection:

`x = 32/25`

The change in x from the given point to the point of intersection:

`Deltax = 32/25 - 8 = -168/25`

The x coordinate of the other end of the line twice the above added to 8:

`x = 8 + 2(-168/25)`

`x = -136/25`

Substitute `x = -136/25` into equation [2]:

`y = 4/3(-136/25) - 5/3`

`y = -544/75 - 125/75`

`y = -669/75 = -223/25`

The other end is the point `(-168/25, -223/25)`