A line segment is bisected by a line with the equation # -7 y + x = 3 #. If one end of the line segment is at #(1 ,6 )#, where is the other end?

Question

A line segment is bisected by a line with the equation #     -7 y  +    x   =   3  #.  If one end of the line segment is at #(1  ,6  )#, where is the other end?

Aurora Ayala · Accepted Answer

The other end of the bisected line could be any point on
#color(white)("XXX")-7y+x=47#

and some other point #color(green)(""(x',y'))#
with an intersection point of #color(red)(-7y+x=3)# and this line segment at #color(red)(""(barx,bary))#

Then
#color(white)("XXX")(6-color(red)(bary))=(color(red)(bary)-color(green)(y'))#
#color(white)("XXXXXX")rarr color(green)(y')=2color(red)(bary)-6#
and
#color(white)("XXX")(1-color(red)(barx))=(color(red)(barx)-color(green)(x'))#
#color(white)("XXXXXX")rarr color(green)(x')=2color(red)(barx)-1#

Specifically, we could solve #color(red)(-7y+x=3)# for a couple arbitrary solution points:
#color(white)("XXX")color(red)((barx_1,bary_1)=(-4,-1))#
and
#color(white)("XXX")color(red)((barx_2,bary_2)=(3,0))#

and obtain end-of-line-segment values:
#color(white)("XXX")color(green)((x'_1,y'_1)=(-9,-8))#
and
#color(white)("XXX")color(green)((x'_2,y'_2)=(5,-6)#

Using the slopes
#color(white)("XXX")(y-color(green)(y'_1))/(x-color(green)(x'_1))=(color(green)(y'_1-y'_2))/(color(green)(x'_1-x'_2))#

#color(white)("XXX")(y-color(green)(""(-8)))/(x-color(green)(""(-9)))=(color(green)(""(-8))-color(green)(""(-6)))/(color(green)(""(-9))-color(green)(""(-5)))#

#color(white)("XXX")(y+8)/(x+9)=1/7#

#color(white)("XXX")7y+56=x+9#

#color(white)("XXX")7y-x=-47#

or (paralleling the given form):
#color(white)("XXX")-7y+x=47#

Sadie Adams · Answer

undefined To find the other end of the line segment bisected by the line with the equation $-7y + x = 3$, given that one end is at (1, 6), follow these steps:

1. Calculate the slope of the given line. Rearrange the equation to solve for $y$ in terms of $x$: $y = \frac{1}{7}x + \frac{3}{7}$. The coefficient of $x$ is the slope of the line.

2. The line segment is bisected by this line, meaning it passes through the midpoint of the line segment. Use the midpoint formula to find the coordinates of the midpoint.

3. Use the midpoint coordinates and the given endpoint (1, 6) to find the equation of the line passing through these two points.

4. Find the intersection point of this line with the line $-7y + x = 3$. This intersection point will be the other end of the line segment.

5. Calculate the coordinates of the intersection point.

These steps will lead to finding the coordinates of the other end of the line segment.