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?

Rewrite the given line in slope-intercept form so that we can obtain the slope, m, of the line.

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

We observe that #m = 7/3#

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

#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 equation of the bisected line is:

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

Subtract equation [2] 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#

To go to the other end of the line we must move twice that far in same direction:

#2Deltax = -378/58#

The add 7 to find the x coordinate of the other end of the line segment:

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

To find the y coordinate of the other end of the line segment substitute 14/28 for x in equation [2]:

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

#y = 313/29#