A line segment is bisected by a line with the equation # 3 y + 5 x = 2 #. If one end of the line segment is at #( 1 , 4 )#, where is the other end?

Question

A line segment is bisected by a line with the equation #    3 y  +  5 x   =   2  #.  If one end of the line segment is at #(     1  ,  4  )#, where is the other end?

Sophia Alfred · Accepted Answer

The other end of the line is at #(-116/34, 46/34)#

In slope-intercept form, write the following line:

#y = -5/3x + 2/3# [1]

The slope, #m = -5/3#

We can determine the slope of the bisected line, n, to be the negative reciprocal of the bisector since it is perpendicular:

#n = -1/m#

#n = 3/5#

Use the given point, #(1, 4)# and the slope #3/5# to find the value of b in the slope-intercept form #y = mx + b#:

#4 = 3/5(1) + b#

#b = 4 - 3/5#

#b = 17/5#

The bisected line's equation is as follows:

#y = 3/5x + 17/5# [2]

Calculate the intersection's x coordinate by deducting equation [1] from equation [2]:

#y - y = 3/5x + 5/3x + 17/5 - 2/3#

#0 =34/15x + 41/15#

#0 =34x + 41#

#-34x = 41#

#x = -41/34#

#Deltax = -41/34 - 1 = -75/34#

The other end of the line's x coordinate is:

#1 + 2Deltax = 1 + 2(-75/34) = -116/34#

To find the corresponding y coordinate, substitute #-116/34# for x in equation 2:

#y = 3/5(-116/34) + 17/5#

#y = 46/34#

Sadie Allen · Answer

undefined To find the other end of the line segment bisected by the line 3y + 5x = 2, we can use the midpoint formula. First, we need to find the coordinates of the midpoint, which is the point of intersection between the given line and the line segment. To find this point, we need to solve the system of equations formed by the given line and the line segment. After finding the coordinates of the midpoint, we can use the midpoint formula to find the coordinates of the other end of the line segment.