# How do you make a perpendicular line when you're given an existing line and its equation? What's the catch?

Nov 15, 2015

See explanation

#### Explanation:

The slope $m$ of a line $L$ that is perpendicular to another line $L '$ is the negative reciprocal of the slope $m '$ of $L '$

$m = - \frac{1}{m} '$

Now, to determine the equation of $L$, we need any point on the line so that we can determine its $y$-intercept

Example:
Find the equation of the line $L$ if it is perpendicular to the line $L '$ with equation $y = 2 x + 1$ and $L$ passes through $\left(1 , - 4\right)$

The slope $m '$ of $L '$ is 2

$\implies m = - \frac{1}{m} ' = - \frac{1}{2}$

Now that we have the slope, let's find the $y$-intercept.
To do this, we substitute the coordinates of a point that the line passes through. In this case, $L$ passes through $\left(1 , - 4\right)$

$y = m x + b$

$\implies - 4 = - \frac{1}{2} \left(1\right) + b$

$\implies - 4 = - \frac{1}{2} + b$

$\implies b = - 4 + \frac{1}{2}$

$\implies b = - \frac{7}{2}$

Hence, the equation of the line $L$ is

$y = - \frac{1}{2} x - \frac{7}{2}$