# 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 ( 8 , 1 ), where is the other end?

Oct 14, 2017

Coordinates of other end points (-6.6,3.6)

#### Explanation:

Assumption : Bisecting line is a perpendicular bisector

Standard form of equation $y = \max + c$
Slope of perpendicular bisector m is given by
$3 y - 7 x = 2$
$y = \left(\frac{7}{3}\right) x + \left(\frac{2}{3}\right)$
$m = \frac{7}{3}$
Slope of line segment is
$y - 1 = - \left(\frac{1}{m}\right) \left(x - 8\right)$
$y - 1 = - \left(\frac{3}{7}\right) \left(x - 8\right)$
$7 y - 7 = 3 x - 24$

7y-3x=-14color(white)((aaaa) Eqn (1)
3y-7x=2color(white)((aaaa) Eqn (2)

Solving Eqns (1) & (2),
$21 y - 9 x = 42$
$21 y - 49 x = 14$
Subtracting and eliminating y term,
$40 x = 28$
$x = \frac{7}{10}$
Substituting value of x in Eqn (1),
$7 y - \left(\frac{21}{10}\right) = 14$
$y = \frac{\frac{161}{10}}{7} = \frac{23}{10}$
Mid point #(7/10,23/10)

Let (x1,y1) the other end point.
$\frac{8 + x 1}{2} = \frac{7}{10}$
$x 1 = - 6.6$
$\frac{1 + y 1}{2} = \frac{23}{10}$
$y 1 = 3.6$

Coordinates of other endpoint $\left(- 6.6 , 3.6\right)$