# How do you write the standard form of the equation of the line that is perpendicular to 6x-4y=-9 and contains (4,-1) ?

Apr 29, 2018

$2 x + 3 y = 5$

#### Explanation:

$\text{the equation of a line in "color(blue)"standard form}$ is.

$\textcolor{red}{\overline{\underline{| \textcolor{w h i t e}{\frac{2}{2}} \textcolor{b l a c k}{A x + B y = C} \textcolor{w h i t e}{\frac{2}{2}} |}}}$

$\text{where A is a positive integer and B, C are integers}$

$\text{obtain the equation in "color(blue)"slope-intercept form}$

•color(white)(x)y=mx+b

$\text{where m is the slope and b the y-intercept}$

$\text{rearrange "6x-4y=-9" into this form}$

$\text{subtract "6x" from both sides}$

$\Rightarrow - 4 y = - 6 x - 9$

$\text{divide all terms by } - 4$

$\Rightarrow y = \frac{3}{2} x + \frac{9}{4} \leftarrow \textcolor{b l u e}{\text{in slope-intercept form}}$

$\text{with slope m } = \frac{3}{2}$

$\text{given a line with slope m then the slope of a line}$
$\text{perpendicular to it is}$

•color(white)(x)m_(color(red)"perpendicular")=-1/m

$\Rightarrow {m}_{\text{perpendicular}} = - \frac{1}{\frac{3}{2}} = - \frac{2}{3}$

$\Rightarrow y = - \frac{2}{3} x + b \leftarrow \textcolor{b l u e}{\text{is the partial equation}}$

$\text{to find b substitute "(4,-1)" into the partial equation}$

$- 1 = - \frac{8}{3} + b \Rightarrow b = - 1 + \frac{8}{3} = \frac{5}{3}$

$\Rightarrow y = - \frac{2}{3} x + \frac{5}{3} \leftarrow \textcolor{b l u e}{\text{in slope intercept form}}$

$\text{multiply all terms by } 3$

$\Rightarrow 3 y = - 2 x + 5$

$\text{add "2x" to both sides}$

$\Rightarrow 2 x + 3 y = 5 \leftarrow \textcolor{red}{\text{in standard form}}$