# How do you find the equation of a line perpendicular to 2x+3y=9 and passing through (3,-3)?

Mar 31, 2018

$3 x - 2 y - 15 = 0$

#### Explanation:

We know that,

the slope of the line $a x + b y + c = 0$ is $m = - \frac{a}{b}$

$\therefore$ The slope of the line $2 x + 3 y = 9$ is ${m}_{1} = - \frac{2}{3}$

$\therefore$ The slope of the line perpendicular to $2 x + 3 y = 9$

is ${m}_{2} = - \frac{1}{m} _ 1 = - \frac{1}{- \frac{2}{3}} = \frac{3}{2}$.

Hence,the equn.of line passing through $\left(3 , - 3\right) \mathmr{and} {m}_{2} = \frac{3}{2}$ is

$y - \left(- 3\right) = \frac{3}{2} \left(x - 3\right)$

$y + 3 = \frac{3}{2} \left(x - 3\right)$

$\implies 2 y + 6 = 3 x - 9$

$\implies 3 x - 2 y - 15 = 0$

Note:

The equn.of line passing through $A \left({x}_{1} , {y}_{1}\right) \mathmr{and}$with slope
$m$ is

$y - {y}_{1} = m \left(x - {x}_{1}\right)$

Mar 31, 2018

$y = \frac{3}{2} x - \frac{15}{2}$

#### Explanation:

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

•color(white)(x)y=mx+b

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

$\text{rearrange "2x+3y=9" into this form}$

$\Rightarrow 3 y = - 2 x + 9$

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

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

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

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

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

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

$\text{to find b substitute "(3,-3)" into the partial equation}$

$- 3 = \frac{9}{2} + b \Rightarrow b = - \frac{6}{2} - \frac{9}{2} = - \frac{15}{2}$

$\Rightarrow y = \frac{3}{2} x - \frac{15}{2} \leftarrow \textcolor{red}{\text{equation of perpendicular line}}$