# What is the equation of the line perpendicular to y - 2x = 3 and passing through the point (2, 5)?

Mar 3, 2018

$y = - \frac{1}{2} x + 6$

#### Explanation:

Given Information
$y - 2 x = 3$ need to find a perpendicular line to this line that passes through the point $\left(2 , 5\right)$

For starters, solve for $y$ in the first equation.

$y - 2 x = 3$
$y = 2 x + 3$

Now that we have the equation that is easy to read, to make a line perpendicular, the slope is ALWAYS the reciprocal. So the slope in this equation is $2 x$. The reciprocal, which is the absolute opposite in sign and numbers (flipping up side down as some call it), would be $- \frac{1}{2} x$. So if you were to graph the lines

$y = 2 x + 3$ and $y = - \frac{1}{2} x + 3$ they would be perpendicular. However, we are not done yet, since we need to get it through the point $\left(2 , 5\right)$.

Doing this, you need to use the equation $y - h = m \left(x - k\right)$. Your teacher may have used this equation, but with different variables.
$y \mathmr{and} x$ will be staying the same so the only variables you need to worry about would be $h , m , \mathmr{and} k$.

$h$ will the the $y$ coordinate of the point you want it to pass through, in this case, that would be $5$, since in $\left(2 , 5\right)$, $5$ is the $y$ coordinate.

$m$ will be the slope, we already calculated that as it's the reciprocal, so it'll be $- \frac{1}{2}$.

$k$ is the $x$ coordinate of the point. In $\left(2 , 5\right)$, the $x$ coordinate is $2$.

So there you have it! Just plug in the known information in the equation.

$y - h = m \left(x - k\right)$ Equation
$y - \left(5\right) = - \frac{1}{2} \left(x - 2\right)$
$y - \left(5\right) = - \frac{1}{2} x + 1$ || Distributed the $- \frac{1}{2}$ to the $x$ and $- 2$ term.

$y = - \frac{1}{2} x + 6$ Added the $5$ from the left side.