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

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

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Answer 1 · 2016-12-18T19:39:05

To other end is at the point $(- \frac{83}{13}, \frac{40}{13})$ $(-83/13,40/13)$

Explanation:

The equation of any line segment that has the line,

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

, as its perpendicular bisector will have the standard form:

$2 x - 3 y = C$ $2x - 3y = C$

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

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

And then solve for C:

$C = - 22$ $C = -22$

The equation of the bisected line segment is:

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

We shall use equations [1] and [2] to find the x coordinate of the point of intersection, $x_{i}$ $x_i$ .

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

$9 x_{i} + 6 y_{i} = 9 [3]$ $9x_i + 6y_i = 9" [3]"$
$4 x_{i} - 6 y_{i} = - 44 [4]$ $4x_i - 6y_i = -44" [4]"$

Add equation [3] to equation [4]:

$13 x_{i} = - 35$ $13x_i = -35$

$x_{i} = - \frac{35}{13}$ $x_i = -35/13$

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:

![Desmos.com]( useruploads.socratic.org )

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

1 Answer

Explanation:

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

1 Answer

Explanation:

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