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?

1 Answer
Douglas K.
Nov 22, 2016

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)

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