A line segment is bisected by a line with the equation # 3 y - 7 x = 2 #. If one end of the line segment is at #(7 ,8 )#, where is the other end?

Question

A line segment is bisected by a line with the equation #   3   y  - 7    x   =   2  #.  If one end of the line segment is at #(7  ,8  )#, where is the other end?

James Atkins · Accepted Answer

The other end is at the point: #(14/29, 313/29)#

To find the line's slope, m, rewrite the provided line in slope-intercept form.

#y = 7/3x + 2/3# #color(white)_[1]#

We observe that #m = 7/3#

The negative reciprocal of m is the bisected line's slope, or n:

#n = -1/m #

#n = -3/7#

Use this slope and the given point to solve for #b# in the slope-intercept form, #y = nx + b#:

#8 = -3/7(7) + b#

#b = 11#

The bisected line's equation is as follows:

#y = -3/7x + 11# #color(white)_[2]#

Equation [2] is subtracted from equation [1]:

#y- y = 7/3x + 3/7x + 2/3 - 11#

#0 = 58/21x - 31/3#

#58/21x = 31/3#

The x coordinate of the point two lines intersect is: #x = 217/58#

The change in x from the point #(7,8)# to the point of intersection is:

#Deltax = 217/58 - 7#

#Deltax = -189/58#

We must travel twice that far in the same direction to reach the other end of the line:

#2Deltax = -378/58#

To determine the other end of the line segment's x coordinate, add 7:

#7 + 2Deltax = 7 -378/58 = 28/58 = 14/29#

In order to determine the y coordinate of the opposite end of the line segment, replace x in equation [2] with 14/28:

#y = -3/7(14/29) + 11#

#y = 313/29#

Wyatt Anderson · Answer

undefined The other end of the line segment is at the point (1, 5).