# A line segment is bisected by a line with the equation  - y + 4 x = 3 . If one end of the line segment is at ( 2 , 6 ), where is the other end?

Oct 6, 2016

Any point on the line $\textcolor{g r e e n}{y = 4 x - 4}$

#### Explanation:

Consider the vertical line $\textcolor{m a \ge n t a}{x = 2}$
Clearly $\textcolor{m a \ge n t a}{x} = 2$ goes through the point color(red)(""(2,6))
and it intersects $\textcolor{b l u e}{- y + 4 x = 3}$ at color(blue)(""(2,5))

color(red)(""(2,6)) is vertically $\textcolor{b r o w n}{1}$ unit above color(blue)(""(2,5)), the intersection point with $\textcolor{b l u e}{- y + 4 x = 3}$

color(green)(""(2,4)) is vertically $\textcolor{b r o w n}{1}$ unit below color(blue)(""(2,5)), the intersection point with $\textcolor{b l u e}{- y + 4 x = 3}$

Therefore $\textcolor{b l u e}{- y + 4 x = 3}$ bisects the line segment between color(green)(""(2,4)) and color(red)(""(2,6))

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Furthermore (as seen in the image below and considering similar triangles)
any point on a line parallel to $\textcolor{b l u e}{- y + 4 x = 3}$ through color(green)(""(2,4)) will provide, together with color(red)(""(2,6)) a line segment bisected by $\textcolor{b l u e}{- y + 4 x = 3}$

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$\textcolor{b l u e}{- y + 4 x = 3}$ can be written in slope-intercept form as
$\textcolor{w h i t e}{\text{XXX}} \textcolor{b l u e}{y = 4 x - 3}$ with a y-intercept at color(blue)(""(-3))

If the line through color(green)(""(2,4)) parallel to $\textcolor{b l u e}{- y + 4 x = 3}$ is $\textcolor{b r o w n}{1}$ unit vertically below $\textcolor{b l u e}{- y + 4 x = 3}$
it will have a y-intercept at color(green)(""(-4))
and therefore a slope-intercept form of
$\textcolor{w h i t e}{\text{XXX}} \textcolor{g r e e n}{y = 4 x - 4}$