A line segment is bisected by a line with the equation # 2 y + 3 x = 3 #. If one end of the line segment is at #( 1 , 8 )#, where is the other end?

1 Answer
Dec 18, 2016

To other end is at the point #(-83/13,40/13)#

Explanation:

The equation of any line segment that has the line,

#3x + 2y = 3" [1]"#

, as its perpendicular bisector will have the standard form:

#2x - 3y = C#

To find the value of C for the specified line segment, substitute the point #(1, 8)# into the equation:

#2(1) - 3(8) = C#

And then solve for C:

#C = -22#

The equation of the bisected line segment is:

#2x - 3y = -22" [2]"#

We shall use equations [1] and [2] to find the x coordinate of the point of intersection, #x_i#.

Multiply equation [1] by 3 and equation [2] by 2:

#9x_i + 6y_i = 9" [3]"#
#4x_i - 6y_i = -44" [4]"#

Add equation [3] to equation [4]:

#13x_i = -35#

#x_i = -35/13#

The change in x (#Deltax) #from the starting point to the point of intersection is as follows:

#Deltax = (-35/13 - 1)#

#Deltax = -48/13#

The x coordinate of the other end, #x_e# can be computed as follows:

#x_e = 2Deltax + x_s#

where #x_s# is the starting x coordinate, #x_s = 1#:

#x_e = 2(-48/13) + 1#

#x_e = -83/13#

To find the corresponding y coordinate, #y_e# substitute #x_e# into equation [2]:

#2(-83/13) - 3y_e = -22#

#y_e = 40/13#

To other end is at the point #(-83/13,40/13)#

Here is a graph with the two lines and the start and end points plotted:

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