A line segment is bisected by a line with the equation # 4 y + 3 x = 4 #. If one end of the line segment is at #( 8 , 9 )#, where is the other end?

1 Answer
Oct 22, 2016

The other end is the point #(-168/25, -223/25)#

Explanation:

Write the equation of the bisector is slope-intercept form:

#y = -3/4x + 1color(white)_[1]#

The slope of the bisected line is the negative reciprocal, #4/3#. Use the point-slope form of the equation of a line to force the line through point #(8,9)#

#y - 9 = 4/3(x - 8)#

#y - 9 = 4/3x - 32/3#

#y = 4/3x - 32/3 + 9#

#y = 4/3x - 5/3color(white)_[2]#

Subtract equation [1] from equation [2]:

#0 = (4/3 + 3/4)x - 8/3#

Solve for x:

#8/3 = 25/12x#

This is the x coordinate of the point of intersection:

#x = 32/25#

The change in x from the given point to the point of intersection:

#Deltax = 32/25 - 8 = -168/25#

The x coordinate of the other end of the line twice the above added to 8:

#x = 8 + 2(-168/25)#

#x = -136/25#

Substitute #x = -136/25# into equation [2]:

#y = 4/3(-136/25) - 5/3#

#y = -544/75 - 125/75#

#y = -669/75 = -223/25#

The other end is the point #(-168/25, -223/25)#