A line segment is bisected by a line with the equation # 4 y - 2 x = 3 #. If one end of the line segment is at #( 5 , 6 )#, where is the other end?

First of all, the bisector of a line segment is perpendicular to the segment.

Perpendicular slope is the negative reciprocal of the original slope, so first step is to find the slope of the given line.

Step 1: Finding the equation of the line segment

Convert to slope-intercept form #y=mx+b#

#4y-2x=3#
#4y = 2x+3#
#y=2/4x+3/4#
#color(red)(m=2/4)#

Taking the negative reciprocal we have the slope of our line segment #color(red)(m=-4/2)#

NOTE: Because the bisector intersects the line segment at the midpoint, we can use the given point to find our #b# value.

#6=-4/2(5)+b#
#6=-20/2+b#
#6+10=b#
#color(red)(b=16)#

This gives us the equation of the line segment on infinite domain #y=-4/2x+16#

Step 2: Equate the two lines to find the intersection point at x

#2/4x+3/4=-4/2x+16#

Group x values together and factor

#x(2/4+4/2)=-3/4+16#
#x(2/4+8/4)=-3/4+64/4#

#(10x)/4=61/4#

#10x=61#

#color(red)(x=6.1)#

Sub in 6.1 to either equation to find y.

#y=-4/2(6.1)+16#
#y=-24.4/2+16#
#y=-12.2+16#
#y=3.8#

So our intersection point is at #(6.1,3.8)#

Step 3: Find the endpoint using the midpoint formula

Because the intersection point #(6.1,3.8)# is also a midpoint, we can find the final point using the midpoint formula.

#((x_1+x_2)/2, (y_1+y_2)/2)#

Substitute the original points at #x_1,y_1#

#(5+x_2)/2=6.1#
#x_2=6.1(2)-5#
#x_2=7.2#

#(6+y_2)/2=3.8#
#y_2=3.8(2)-6#
#y_2=1.6#

Answer

#(x,y)=(7.2,1.6)#