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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