Question

A line segment is bisected by a line with the equation `3 y + 5 x = 2`. If one end of the line segment is at `( 5 , 8 )`, where is the other end?

Answer 1

The other end is at `(-150/17, -5/17)`

Explanation:

This problem is in the perpendicular bisectors so it is safe to assume that the bisected line is perpendicular; that means that its equation is of the form:

`5y - 3x = c`

To find the value of c, substitute in the point `(5, 8)`

`5(8) - 3(5) = c`

`c = 25`

The equation of the bisected line is:

`5y - 3x = 25" [1]"`

The given equation is:

`3y + 5x = 2" [2]"`

Multiply equation [1] by -3 and equation [2] by 5

`-15y + 9x = -75" [3]"`
`15y + 25x = 10" [4]"`

Add equations [3] and [4]:

`34x = -65`

`x = -65/34`

Let `Deltax` = the change in x from the given point to the intersection point:

`Deltax = -65/34 - 5 = -235/34`

The x coordinate of the other end of the line segment is twice that change added to the starting x coordinate:

`x_(end) = 2Deltax + 5`

`x_(end) = 2(-235/34) + 5`

`x_(end) = -150/17`

To find the y coordinate, `y_(end)`, substitute `-150/17` for x

`5y - 3(-150/17) = 25`

`5y = 3(-150/17) + 25`

`y_(end) = 3(-30/17) + 5`

`y_(end) = -5/17`

The other end is at `(-150/17, -5/17)`