# What is the equation of the line perpendicular to y=1/4x  that passes through  (-7,4) ?

Apr 21, 2017

$y = - 4 x - 24$

#### Explanation:

$y = \frac{1}{4} x \text{ is in "color(blue)"slope-intercept form}$ that is.

$\textcolor{red}{\overline{\underline{| \textcolor{w h i t e}{\frac{2}{2}} \textcolor{b l a c k}{y = m x + b} \textcolor{w h i t e}{\frac{2}{2}} |}}}$
where m represents the slope and b, the y-intercept.

$\Rightarrow y = \frac{1}{4} x \text{ has slope } = m = \frac{1}{4}$

The slope of a line perpendicular to this is $\textcolor{b l u e}{\text{the negative reciprocal}}$ of m

$\Rightarrow {m}_{\text{perpendicular}} = - \frac{1}{\frac{1}{4}} = - 4$

The equation of a line in $\textcolor{b l u e}{\text{point-slope form}}$ is.

$\textcolor{red}{\overline{\underline{| \textcolor{w h i t e}{\frac{2}{2}} \textcolor{b l a c k}{y - {y}_{1} = m \left(x - {x}_{1}\right)} \textcolor{w h i t e}{\frac{2}{2}} |}}}$
where $\left({x}_{1} , {y}_{1}\right) \text{ is a point on the line}$

$\text{using " m=-4" and } \left({x}_{1} , {y}_{1}\right) = \left(- 7 , 4\right)$

$y - 4 = - 4 \left(x - \left(- 7\right)\right)$

$\Rightarrow y - 4 = - 4 \left(x + 7\right) \leftarrow \textcolor{red}{\text{ in point-slope form}}$

$\text{distributing and simplifying gives}$

$y - 4 = - 4 x - 28$

$\Rightarrow y = - 4 x - 24 \leftarrow \textcolor{red}{\text{ in slope-intercept form}}$