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?

1 Answer
Oct 24, 2016

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#