# How do you write an equation in slope intercept form perpendicular to y=1/2x-4, contains (4,1)?

Mar 19, 2017

$y = - 2 x + 9$

#### Explanation:

An equation in the form $y = \textcolor{g r e e n}{m} x + \textcolor{b l u e}{b}$ is in slope intercept form with a slope of $\textcolor{g r e e n}{m}$

Therefore $y = \textcolor{g r e e n}{\frac{1}{2}} x - 4$ has a slope of $\textcolor{g r e e n}{\frac{1}{2}}$

If a line has a slope of $\textcolor{g r e e n}{m}$ then any line perpendicular to it has a slope of $\textcolor{m a \ge n t a}{- \frac{1}{m}}$

Therefore any line perpendicular to $y = \textcolor{g r e e n}{\frac{1}{2}} x - 4$
must have a slope of $\textcolor{m a \ge n t a}{- 2}$
and a slope-intercept form of $y = \left(\textcolor{m a \ge n t a}{- 2}\right) x + \textcolor{b l u e}{b}$ for some constant $\textcolor{b l u e}{b}$

If $\left(\textcolor{red}{x} , \textcolor{\mathmr{and} a n \ge}{y}\right) = \left(\textcolor{red}{4} , \textcolor{\mathmr{and} a n \ge}{1}\right)$ is to be a point on the required perpendicular line then
$\textcolor{w h i t e}{\text{XXX}} \textcolor{\mathmr{and} a n \ge}{1} = \left(\textcolor{m a \ge n t a}{- 2}\right) \cdot \textcolor{red}{4} + \textcolor{b l u e}{b}$

$\textcolor{w h i t e}{\text{XXX}} \rightarrow \textcolor{b l u e}{b} = 9$

and the equation of the required perpendicular line is
$\textcolor{w h i t e}{\text{XXX}} y = \textcolor{m a \ge n t a}{- 2} x + \textcolor{b l u e}{9}$

Mar 19, 2017

$y = - 2 x + 9$

#### Explanation:

$y = \frac{1}{2} x - 4 \text{ is in "color(blue)"slope-intercept form}$

$\text{That is " y=color(red)(m)xcolor(blue)(+b) " where } \textcolor{red}{m}$ represents the slope and $\textcolor{b l u e}{+ b}$ the y-intercept.

$\Rightarrow \text{ slope } = m = \frac{1}{2}$

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

$\Rightarrow {m}_{\text{perpendicular}} = - \frac{1}{\frac{1}{2}} = - 2$

The equation can be written partially as $y = - 2 x + b$

To calculate b substitute (4 ,1) into the partial equation.

$\Rightarrow \left(- 2 \times 4\right) + b = 1$

$\Rightarrow b = 1 + 8 = 9$

$\Rightarrow y = - 2 x + 9 \leftarrow \textcolor{red}{\text{ in slope-intercept form}}$