# A line segment is bisected by a line with the equation  - 3 y + 2 x = 2 . If one end of the line segment is at ( 7 , 9 ), where is the other end?

Jun 18, 2017

(151/13,27/13)

#### Explanation:

Rewrite the equation of the bisector in the form $y = m x + b$
$y = \frac{2}{3} x - \frac{2}{3}$
Therefore its gradient equals to $\frac{2}{3}$.
Since the gradient product of two perpendicular lines is $- 1$, the slope of the unknown line segment should be $\frac{- 1}{\frac{2}{3}} = - \frac{3}{2}$

Solve for the equation of the line segment (which includes the point $\left(7 , 9\right)$. $y = - \frac{3}{2} \left(x - 7\right) + 9$, simplify (optional) and you get $y = - \frac{3}{2} x + \frac{39}{2}$

Solve for the $x$-coordinate for the intersection of the two perpendicular lines: Let $- \frac{3}{2} \left(x - 7\right) + 9 = \frac{2}{3} x - \frac{2}{3} \text{, } x = \frac{121}{13}$

Now you can find the $x$-coordinate of the other end of the segment. Using the formula ${x}_{1} - x = - \left({x}_{2} - x\right)$, where $x$ is the $x$-coordinate of the point in the middle. ${x}_{1} = 7$ and ${x}_{2} = 2 \cdot x - {x}_{1}$, ${x}_{2} = \frac{151}{13}$

Solve for the $y$-coordinate using the equation for the line segment. ${y}_{2} = - \frac{3}{2} \left(x - 7\right) + 9 = \frac{27}{13}$
Therefore coordinate for the other end point is $\left(\frac{151}{13} , \frac{27}{13}\right)$

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