# 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?

Dec 18, 2016

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

#### Explanation:

The equation of any line segment that has the line,

$3 x + 2 y = 3 \text{ }$

, as its perpendicular bisector will have the standard form:

$2 x - 3 y = C$

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

$2 \left(1\right) - 3 \left(8\right) = C$

And then solve for C:

$C = - 22$

The equation of the bisected line segment is:

$2 x - 3 y = - 22 \text{ }$

We shall use equations  and  to find the x coordinate of the point of intersection, ${x}_{i}$.

Multiply equation  by 3 and equation  by 2:

$9 {x}_{i} + 6 {y}_{i} = 9 \text{ }$
$4 {x}_{i} - 6 {y}_{i} = - 44 \text{ }$

Add equation  to equation :

$13 {x}_{i} = - 35$

${x}_{i} = - \frac{35}{13}$

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

$\Delta x = \left(- \frac{35}{13} - 1\right)$

$\Delta x = - \frac{48}{13}$

The x coordinate of the other end, ${x}_{e}$ can be computed as follows:

${x}_{e} = 2 \Delta x + {x}_{s}$

where ${x}_{s}$ is the starting x coordinate, ${x}_{s} = 1$:

${x}_{e} = 2 \left(- \frac{48}{13}\right) + 1$

${x}_{e} = - \frac{83}{13}$

To find the corresponding y coordinate, ${y}_{e}$ substitute ${x}_{e}$ into equation :

$2 \left(- \frac{83}{13}\right) - 3 {y}_{e} = - 22$

${y}_{e} = \frac{40}{13}$

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

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