# A line segment is bisected by a line with the equation  - 4 y + x = 1 . If one end of the line segment is at ( 7 , 2 ), where is the other end?

Nov 30, 2016

The other end is at $\left(\frac{123}{17} , \frac{18}{17}\right)$

#### Explanation:

Let's write the given line in the form $a x + b y = c$

$x - 4 y = 1 \text{ }$

The general equation of lines perpendicular to this line is:

$4 x + y = c$

To find the value of c, substitute 7 for x and 2 for y:

$4 \left(7\right) + 2 = c$

$c = 30$

The equation of the bisected line segment is:

$4 x + y = 30 \text{ }$

The midpoint is at the intersection of of these two lines:

$x - 4 y = 1 \text{ }$
$4 x + y = 30 \text{ }$

Multiply equation  by 4 and subtract from equation 

$17 y = 26$

$y = \frac{26}{17}$

Let $\Delta y =$ the change in the y coordinate = $\frac{26}{17} - 2$

The y coordinate of the other end of the line, ${y}_{1}$ will have 2 twice the change from the starting y coordinate:

${y}_{1} = 2 \Delta y + {y}_{0}$

${y}_{1} = 2 \left(\frac{26}{17} - 2\right) + 2$

${y}_{1} = \frac{52}{17} - 2$

${y}_{1} = \frac{18}{17}$

To find the corresponding x coordinate, substitute $\frac{18}{17}$ for y into equation :

$4 x + \frac{18}{17} = 30$

$4 x = 30 - \frac{18}{17}$

${x}_{1} = \frac{123}{17}$

The other end is at $\left(\frac{123}{17} , \frac{18}{17}\right)$

Here is a graph of two lines and two points: 