# A line segment is bisected by a line with the equation  9 y + x = 5 . If one end of the line segment is at ( 7 , 4 ), where is the other end?

Jun 15, 2017

The other end is: $\left(\frac{249}{41} , - \frac{178}{41}\right)$
Here is a graph:

#### Explanation:

Given: $x + 9 y = 5 \text{ [1]}$

The family of perpendicular lines is:

$9 x - y = k$

Use the point $\left(7 , 4\right)$ to find the value of x:

$9 \left(7\right) - 4 = k$

$k = 59$

The equation of the bisected line is:

$9 x - y = 59 \text{ [2]}$

Multiply equation [2] by 9 and add to equation [1]:

$82 x = 536$

${x}_{\text{midpoint}} = \frac{268}{41}$

Use the midpoint equation to find ${x}_{\text{end}}$

x_"midpoint" = (x_"start"+x_"end")/2

$\frac{268}{41} = \frac{7 + {x}_{\text{end}}}{2}$

${x}_{\text{end}} = \frac{536}{41} - 7$

${x}_{\text{end}} = \frac{249}{41}$

Find ${y}_{\text{end}}$ by substituting ${x}_{\text{end}}$ into equation [2]:

$9 \left(\frac{249}{41}\right) - {y}_{\text{end}} = 59$

${y}_{\text{end}} = 9 \left(\frac{249}{41}\right) - 59$

${y}_{\text{end}} = - \frac{178}{41}$