# A line segment is bisected by a line with the equation  -3 y + x = 1 . If one end of the line segment is at (1 ,6 ), where is the other end?

Oct 26, 2016

The other end is a $\left(4.6 , - 4.8\right)$

#### Explanation:

Write the equation for the bisector in slope-intercept form:

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

The slope is, $m = \frac{1}{3}$

The slope, n, for the bisected line is, $n = - \frac{1}{m} = - \frac{1}{\frac{1}{3}} = - 3$

Use the slope and the point, $\left(1 , 6\right)$ into the slope, $- 3$, into the slope-intercept form of a line and then solve for b:

$6 = - 3 \left(1\right) + b$

$b = 9$

The equation for the bisected line is:

$y = - 3 x + 9$ $\left[2\right]$

Subtract equation $\left[2\right]$ from equation $\left[1\right]$

$y - y = \frac{1}{3} x + 3 x - \frac{1}{3} - 9$

$0 = \frac{10}{3} x - \frac{28}{3}$

The x coordinate of the point of intersection is:

$x = 2.8$

To go from 1 to 2.8, the x coordinate increased 1.8, therefore, to go to the other end of the line x coordinate must increase twice that much, 3.6.

$x = 1 + 3.6 = 4.6$

This is the x coordinate of the other end of the line.

To find the y coordinate, substitute 4.6 for x into equation $\left[2\right]$

#y = -3(4.6) + 9

$y = - 4.8$