# A line segment is bisected by a line with the equation  3 y - 7 x = 2 . If one end of the line segment is at (7 ,8 ), where is the other end?

Oct 28, 2016

The other end is at the point: $\left(\frac{14}{29} , \frac{313}{29}\right)$

#### Explanation:

Rewrite the given line in slope-intercept form so that we can obtain the slope, m, of the line.

$y = \frac{7}{3} x + \frac{2}{3}$ $\textcolor{w h i t e}{_} \left[1\right]$

We observe that $m = \frac{7}{3}$

The slope, n, of the bisected line is the negative reciprocal of m:

$n = - \frac{1}{m}$

$n = - \frac{3}{7}$

Use this slope and the given point to solve for $b$ in the slope-intercept form, $y = n x + b$:

$8 = - \frac{3}{7} \left(7\right) + b$

$b = 11$

The equation of the bisected line is:

$y = - \frac{3}{7} x + 11$ $\textcolor{w h i t e}{_} \left[2\right]$

Subtract equation  from equation :

$y - y = \frac{7}{3} x + \frac{3}{7} x + \frac{2}{3} - 11$

$0 = \frac{58}{21} x - \frac{31}{3}$

$\frac{58}{21} x = \frac{31}{3}$

The x coordinate of the point two lines intersect is: $x = \frac{217}{58}$

The change in x from the point $\left(7 , 8\right)$ to the point of intersection is:

$\Delta x = \frac{217}{58} - 7$

$\Delta x = - \frac{189}{58}$

To go to the other end of the line we must move twice that far in same direction:

$2 \Delta x = - \frac{378}{58}$

The add 7 to find the x coordinate of the other end of the line segment:

$7 + 2 \Delta x = 7 - \frac{378}{58} = \frac{28}{58} = \frac{14}{29}$

To find the y coordinate of the other end of the line segment substitute 14/28 for x in equation :

$y = - \frac{3}{7} \left(\frac{14}{29}\right) + 11$

$y = \frac{313}{29}$