# A line segment is bisected by a line with the equation  3 y + 5 x = 2 . If one end of the line segment is at ( 1 , 4 ), where is the other end?

Nov 4, 2016

The other end of the line is at $\left(- \frac{116}{34} , \frac{46}{34}\right)$

#### Explanation:

Write the given line in slope-intercept form:

$y = - \frac{5}{3} x + \frac{2}{3}$ [1]

The slope, $m = - \frac{5}{3}$

Because the bisected line is perpendicular, we know that its slope, n, is the negative reciprocal of the bisector:

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

$n = \frac{3}{5}$

Use the given point, $\left(1 , 4\right)$ and the slope $\frac{3}{5}$ to find the value of b in the slope-intercept form $y = m x + b$:

$4 = \frac{3}{5} \left(1\right) + b$

$b = 4 - \frac{3}{5}$

$b = \frac{17}{5}$

The equation of the bisected line is:

$y = \frac{3}{5} x + \frac{17}{5}$ [2]

Find the x coordinate of the intersection by subtracting equation [1] from equation [2]:

$y - y = \frac{3}{5} x + \frac{5}{3} x + \frac{17}{5} - \frac{2}{3}$

$0 = \frac{34}{15} x + \frac{41}{15}$

$0 = 34 x + 41$

$- 34 x = 41$

$x = - \frac{41}{34}$

$\Delta x = - \frac{41}{34} - 1 = - \frac{75}{34}$

The x coordinate of the other end of the line is:

$1 + 2 \Delta x = 1 + 2 \left(- \frac{75}{34}\right) = - \frac{116}{34}$

To find the corresponding y coordinate, substitute $- \frac{116}{34}$ for x in equation 2:

$y = \frac{3}{5} \left(- \frac{116}{34}\right) + \frac{17}{5}$

$y = \frac{46}{34}$