# How do you find the general form of the line perpendicular to 3x+5y-8=0 that passes through the point (-8,1)?

May 21, 2018

Therefore equation of the perpendicular line is:

$y = \frac{5}{3} x - \frac{37}{3}$ #### Explanation:

Slope of perpendicular line is negative reciprocal of the original slope of the line.

So given linear equation is:
$3 x + 5 y - 8 = 0$ -----> re-write this equation as $y = m x + b$ where $m$ is the slope of the line and $b$ is the y-intercept.

$5 y = 8 - 3 x$

$5 y = - 3 x + 8$

$y = - \frac{3}{5} y + \frac{8}{5}$ -----> Slope of the line is $- \frac{3}{5}$ ----> $\textcolor{red}{R E D - G R A P H}$

So the slope of the perpendicular line is $- \frac{1}{- \frac{3}{5}}$ = $\frac{5}{3}$

So the equation of the perpendicular line is:
$y = \frac{5}{3} x + b$

Lets us find $b$ from the given points (-8,1).

$y = \frac{5}{3} x + b$

$1 = \left(\frac{5}{3} \times 8\right) + b$

$1 = \frac{40}{3} + b$

$b = 1 - \frac{40}{3}$

$b = - \frac{37}{3}$

Therefore equation of the perpendicular line is:

$y = \frac{5}{3} x - \frac{37}{3}$ ----> $\textcolor{b l u e}{B L U E - G R A P H}$

We can draw the graph and check as well (see the attached graphs)

May 21, 2018

$5 x - 3 y + 43 = 0$

#### Explanation:

$\text{the general form of the equation of a line is}$

$\textcolor{red}{\overline{\underline{| \textcolor{w h i t e}{\frac{2}{2}} \textcolor{b l a c k}{A x + B y + C = 0} \textcolor{w h i t e}{\frac{2}{2}} |}}}$

$\text{where A, B and C are integers with A and B non-zero}$

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

•color(white)(x)y=mx+b

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

$\text{rearrange "3x+5y-8=0" into this form}$

$\text{subtract "3x-8" from both sides}$

$\Rightarrow 5 y = - 3 x + 8$

$\text{divide all terms by 5}$

$\Rightarrow y = - \frac{3}{5} x + \frac{8}{5} \leftarrow \textcolor{b l u e}{\text{in slope-intercept form}}$

$\text{with slope m } = - \frac{3}{5}$

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

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

$\Rightarrow {m}_{\text{perpendicular}} = - \frac{1}{- \frac{3}{5}} = \frac{5}{3}$

$\text{now find the equation of the perpendicular line}$

$\Rightarrow y = \frac{5}{3} x + b \leftarrow \textcolor{b l u e}{\text{is the partial equation}}$

$\text{to find b substitute "(-8,1)" into the partial equation}$

$1 = - \frac{40}{3} + b \Rightarrow b = \frac{3}{3} + \frac{40}{3} = \frac{43}{3}$

$\Rightarrow y = \frac{5}{3} x + \frac{43}{3} \leftarrow \textcolor{red}{\text{in slope-intercept form}}$

$\text{rearrange into general form by multiplying all terms by 3}$

$\Rightarrow 3 y = 5 x + 43$

$\text{subtract "3y" from both sides}$

$\Rightarrow 5 x - 3 y + 43 = 0 \leftarrow \textcolor{red}{\text{in general form}}$