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

Oct 5, 2016

The other end will be any point on the line $4 y - 3 x = - 10$

Explanation:

Consider the vertical line $x = 2$ which passes through $\left(2 , 5\right)$

$x = 2$ will intersect $4 y - 3 x = 2$ at $\left(2 , 2\right)$

$\left(2 , 5\right)$ is $3$ units vertically above $\left(2 , 2\right)$;
that is $\left(2 , 5\right)$ is vertically $3$ units above $4 y - 3 x = 2$

Any point $3$ units vertically below $4 y - 3 x = 2$ will provide a second point which together with $\left(2 , 5\right)$ form a line segment bisected by $4 y - 3 x = 2$

Re-writing $4 y - 3 x = 2$ in slope-intercept form: $y = \frac{3}{4} x + \frac{2}{4}$

The y-intercept for a line $3$ units below $3 = \frac{3}{4} x + \frac{2}{4}$ will be at $\frac{2}{4} - 3 = - \frac{10}{4}$

Therefore this second line will have a slope-intercept equation of
$\textcolor{w h i t e}{\text{XXX}} y = \frac{3}{4} x - \frac{10}{4}$
or in a form similar to the initial equation:
$\textcolor{w h i t e}{\text{XXX}} 4 y - 3 x = - 10$

The image below may help: