# How do you write an equation of a line passing through (2, 6), perpendicular to 2x -3y = 12?

Jan 21, 2017

$3 x + 2 y = 18$

#### Explanation:

$\textcolor{red}{\text{THINGS TO REMEMBER}}$

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$\textcolor{b l u e}{\text{If a line has a slope of } m}$
$\textcolor{w h i t e}{\text{XXXX")color(blue)("all lines perpendicular to it have a slope of } - \frac{1}{m}}$

color(red)(bar(color(white)("XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX")

$\textcolor{w h i t e}{\text{XX")color(blue)("A line in standard form: "Ax+BY=C)color(white)("XX}}$
$\textcolor{w h i t e}{\text{XXXX")color(blue)("has a slope of } - \frac{A}{B}}$

color(red)(bar(color(white)("XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX")

$\textcolor{w h i t e}{\text{XX")color(blue)("The point-slope form for a line with}}$
color(white)("XX")color(blue)("point "(a,b)" and slope "m" is"color(white)(*XXX"))
$\textcolor{w h i t e}{\text{XXXXXX}} \textcolor{b l u e}{y - a = m \left(x - b\right)}$

color(red)(bar(color(white)("XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX")

$\textcolor{g r e e n}{\text{SOLVING THE GIVEN PROBLEM}}$

$2 x - 3 y = 12$ has a slope of $m = \frac{2}{3}$

All lines perpendicular to $2 x - 3 y = 12$ have a slope of $- \frac{3}{2}$

The equation of a line with slope $\left(- \frac{3}{2}\right)$ through the point $\left(2 , 6\right)$
in slope-point form is:
$\textcolor{w h i t e}{\text{XXXXXX}} y - 6 = - \frac{3}{2} \left(x - 2\right)$

$\textcolor{g r e e n}{\text{~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~}}$

While the above slope-point form would be a valid answer,
we would normally convert this into standard form:

$y - 6 = - \frac{3}{2} \left(x - 2\right)$

$\textcolor{w h i t e}{\text{XXX}} \rightarrow 2 y - 12 = - 3 x + 6$

$\textcolor{w h i t e}{\text{XXX}} \rightarrow 3 x + 2 y = 18$

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