# How do you write the equation that represents the line perpendicular to y=-3x +4 and passing through the point (-1,1)?

Dec 12, 2016

$y = \frac{1}{3} x + \frac{4}{3}$

#### Explanation:

Step 1) Because this is already solved for $y$ we can get the slope of the perpendicular line by taking the slope of the given line (-3) and "flipping" it and reversing the sign:

$- 3$ becomes $- - \frac{1}{3} \to + \frac{1}{3} \to \frac{1}{3}$

Therefore the slope of the perpendicular line is $\frac{1}{3}$

Step 2) Use the point slope for to find the perpendicular equation:

$y - 1 = \frac{1}{3} \left(x - - 1\right)$

$y - 1 = \frac{1}{3} \left(x + 1\right)$

$y - 1 = \frac{1}{3} x + \frac{1}{3}$

$y - 1 + 1 = \frac{1}{3} x + \frac{1}{3} + 1$

$y - 0 = \frac{1}{3} x + \frac{1}{3} + \frac{3}{3}$

$y = \frac{1}{3} x + \frac{4}{3}$

Dec 12, 2016

$y = \left(\frac{1}{3}\right) x + \left(\frac{4}{3}\right)$

#### Explanation:

Any linear equation of the form
$\textcolor{w h i t e}{\text{XXX}} y = \textcolor{g r e e n}{m} x + \textcolor{b l u e}{b}$
has a slope of $\textcolor{g r e e n}{m}$

Any line perpendicular to a line with slope $\textcolor{g r e e n}{m}$
has a slope color(green)(""(-1/m))

$y = \textcolor{g r e e n}{- 3} x + 4$ has a slope of $\textcolor{g r e e n}{- 3}$
$\rightarrow$ any line perpendicular to it has a slope $\textcolor{g r e e n}{\frac{1}{3}}$

A line with slope $\textcolor{m a \ge n t a}{k}$ through the point $\left(\textcolor{red}{a} , \textcolor{b l u e}{b}\right)$
can be written in slope-point form as:
$\textcolor{w h i t e}{\text{XXX}} \left(y - \textcolor{b l u e}{b}\right) = \textcolor{m a \ge n t a}{k} \left(x - \textcolor{red}{a}\right)$

Therefore the line perpendicular to $y = - 3 x + 4$ and through the point $\left(\textcolor{red}{- 1} , \textcolor{b l u e}{1}\right)$
can be written as:
color(white)("XXX")y-color(red)1=color(green)(1/3)(x-color(red)(""(-1)))

While this is a perfectly valid answer to the given question, we would normally convert this into standard form as:
$\textcolor{w h i t e}{\text{XXX}} x - 3 y = - 4$
or slope-vertex form (as was the initial equation):
$\textcolor{w h i t e}{\text{XXX}} y = \frac{1}{3} x + \frac{4}{3}$