# A line segment is bisected by a line with the equation  2 y + 9 x = 3 . If one end of the line segment is at (3 ,2 ), where is the other end?

Nov 27, 2016

The point $\left(- \frac{249}{85} , \frac{58}{85}\right)$

#### Explanation:

For a line with a slope that is perpendicular to the given line

$9 x + 2 y = 3 \text{ }$

Swap the coefficients of x and y, change the sign of one of the coefficients, and set it equal to an arbitrary constant:

$2 x - 9 y = C$

To find the value of the constant, substitute the point $\left(3 , 2\right)$ into the equation:

$2 \left(3\right) - 9 \left(2\right) = C$

$C = - 12$

The equation of the bisected line is:

$2 x - 9 y = - 12 \text{ }$

Multiply equation  by 9 and equation  by 2:

$81 x + 18 y = 27 \text{ }$
$4 x - 18 y = - 24 \text{ }$

Add equation  to equation :

$85 x = 3$

$x = \frac{3}{85}$

This is the x coordinate of intersection.

Let $\Delta x =$ the change from the original x coordinate, 3, to the x coordinate of intersection:

$\Delta x = \left(\frac{3}{85} - 3\right) = - \frac{252}{85}$

Let ${x}_{1} =$ the x coordinate of the other end of the line segment.

${x}_{1} = 2 \Delta x + 3$

${x}_{1} = - \frac{504}{85} + 3$

${x}_{1} = - \frac{249}{85}$

To find the corresponding y coordinate, substitute the value of ${x}_{1}$ into equation :

$2 \left(- \frac{249}{85}\right) - 9 {y}_{1} = - 12$

${y}_{1} = \frac{58}{85}$