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

Sep 6, 2016

Find the equation of a line that is perpendicular to the give line and goes through (7,2), then use the midpoint formula.

#### Explanation:

Find the slope of $- y + 7 x = 1$ by changing it to slope-intercept form:
$y = 7 x - 1$, $m = 7$

The slope of a line perpendicular to this is the negative reciprocal of 7 or $- \frac{1}{7}$, $m = - \frac{1}{7}$

Use the point slope formula to find the equation of the line perpendicular to the given line that goes through (7,2). This is the equation of the line that includes the line segment.
$y - {y}_{1} = m \left(x - {x}_{1}\right)$

$y - 2 = - \frac{1}{7} \left(x - 7\right)$
$y - 2 = - \frac{1}{7} x + 1$
$y = - \frac{1}{7} x + 3$, or $\frac{1}{7} x + y = 3$

Now find the intersection of the given line with the line that contains the line segment, by setting up the two equations as a system of equations and solving.
$\frac{1}{7} x + y = 3$
$7 x - y = 1$

Adding these two equations together gives $\frac{50}{7} x = 4$ and $x = \frac{14}{25}$

Plug this value of x into either of the equations to find y. The solution to the system is the intersection of the two lines and is $\left(\frac{14}{25} , \frac{73}{25}\right)$

The intersection of the two lines is the midpoint of the segment, because the perpendicular bisector of a segment goes through the midpoint of the segment. Now use the midpoint formula to find the other endpoint of the segment.

Midpoint $\left(x , y\right) = \left(\frac{{x}_{1} + {x}_{2}}{2} , \frac{{y}_{1} + {y}_{2}}{2}\right)$

$\left(\frac{14}{25} , \frac{73}{25}\right) = \left(\frac{7 + {x}_{2}}{2} , \frac{2 + {y}_{2}}{2}\right)$

$\frac{14}{25} = \frac{7 + {x}_{2}}{2}$ and $\frac{73}{25} = \frac{2 + {y}_{2}}{2}$

Cross multiply and solve.
$\left({x}_{2} , {y}_{2}\right) = \left(- \frac{147}{25} , \frac{96}{25}\right)$