A line segment is bisected by a line with the equation # - 2 y + 3 x = 1 #. If one end of the line segment is at #(6 ,3 )#, where is the other end?

Question

A line segment is bisected by a line with the equation #   -  2 y  +  3  x   =   1  #.  If one end of the line segment is at #(6  ,3  )#, where is the other end?

Noah Atkinson · Accepted Answer

#(12/13,83/13)#

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

If the line segment was a line perpendicular to #[1]#. then its gradient would be the negative reciprocal of the gradient of #[1]#

#-2/3#

Creating a line equation for this:

#y-3=-2/3(x-6)#

#y=-2/3x+7\ \ \ \ [2]#

Finding the intersection of #[1] and [2]#

#-2/3x+7=3/2x-1/2#

#x=45/13#

Substituting in #[1]#

#y=3/2(45/13)-1/2=61/13#

The midpoint's coordinates are provided by:

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

We have #(6,3)# #:.#

#((6+x_2)/2,(3+y_2)/2)=>(45/13,61/13)#

Hence:

#(6+x)/2=45/13=>x=12/13#

#(3+y)/2=61/13=>y=83/13#

The opposite end's coordinates are:

#(12/13,83/13)#

Mason Adams · Answer

undefined To find the other end of the line segment bisected by the line $ -2y + 3x = 1 $ with one end at (6, 3), follow these steps:

1. Find the slope of the line $ -2y + 3x = 1 $.
2. Find the equation of the perpendicular bisector to this line passing through the given point (6, 3).
3. Find the intersection point of the perpendicular bisector with the given line to determine the other end of the line segment.

Step 1:
The equation of the given line is $ -2y + 3x = 1 $. Rewrite it in slope-intercept form: $ y = \frac{3}{2}x - \frac{1}{2} $. The slope of this line is $ \frac{3}{2} $.

Step 2:
Since the line segment is bisected, the slope of the perpendicular bisector is the negative reciprocal of the slope of the given line, which is $ -\frac{2}{3} $. Using the point-slope form of a line, the equation of the perpendicular bisector passing through the point (6, 3) is:
$$ y - 3 = -\frac{2}{3}(x - 6) $$

Step 3:
To find the intersection point of the perpendicular bisector with the given line, solve the system of equations formed by the equations of both lines:
$$ \begin{cases} y = \frac{3}{2}x - \frac{1}{2} \ y - 3 = -\frac{2}{3}(x - 6) \end{cases} $$
$$ \begin{cases} y = \frac{3}{2}x - \frac{1}{2} \ y = -\frac{2}{3}x + 5 \end{cases} $$
Solve this system to find the coordinates of the other end of the line segment.

The solution is $ x = \frac{4}{5} $ and $ y = \frac{17}{5} $. So, the other end of the line segment is at the point $ \left(\frac{4}{5}, \frac{17}{5}\right) $.

Sadie Allen · Answer

undefined To find the other end of the line segment, we need to find the coordinates where the line -2y + 3x = 1 intersects the line segment. Since the line segment is bisected, it means the midpoint of the line segment lies on the given line.

First, find the midpoint of the line segment using the coordinates (6, 3) and the midpoint formula.

Midpoint formula:
$$ \text{Midpoint} = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) $$

Substitute the given point (6, 3) as (x₁, y₁) and solve for the other endpoint.
$$ x₁ = 6, \, y₁ = 3 $$

Then, substitute these values into the equation of the line -2y + 3x = 1 and solve for x₂ and y₂.

$$ -2y + 3x = 1 $$
$$ -2(3) + 3(6) = 1 $$
$$ -6 + 18 = 1 $$
$$ 12 = 1 $$

The equation has no solutions, which indicates there is no intersection between the given line and the line segment. Therefore, the other end of the line segment cannot be determined based on the given information.