# A line segment is bisected by a line with the equation  -3 y + 7 x = 1 . If one end of the line segment is at (9 ,2 ), where is the other end?

Oct 24, 2016

The point $\left(- \frac{262}{58} , \frac{452}{58}\right)$

#### Explanation:

Write the given equation of the bisector line in slope-intercept:

$y = \frac{7}{3} x - \frac{1}{3}$ [1]

The slope of the line that goes through point $\left(9 , 2\right)$ is the negative reciprocal of the slope of the bisector line, $- \frac{3}{7}$. Use the point-slope form of the equation of a line to write the equation for the line segment:

$y - 2 = - \frac{3}{7} \left(x - 9\right)$

$y - 2 = - \frac{3}{7} x + \frac{27}{7}$

$y = - \frac{3}{7} x + \frac{27}{7} + 2$

$y = - \frac{3}{7} x + \frac{41}{7}$ [2]

To find the x coordinate of the point of intersection subtract equation [2] from equation [1]

$y - y = \frac{7}{3} x + \frac{3}{7} x - \frac{1}{3} - \frac{41}{7}$

$0 = \frac{58}{21} x - \frac{130}{21}$

$x = \frac{130}{58}$

The change in x from 9 to $\frac{214}{58}$ is:

$\Delta x = \frac{130}{58} - 9$

$\Delta x = - \frac{392}{58}$

The x coordinate for other end will have twice that change:

$x = 9 - \frac{784}{58}$

$x = - \frac{262}{58}$

To find the y coordinate of the other end, substitute the x coordinate into equation [2]

$y = - \frac{3}{7} \left(- \frac{262}{58}\right) + \frac{41}{7}$

$y = \frac{452}{58}$