# What is the equation of the line perpendicular to y=6x  that passes through  (6,-1) ?

Jul 20, 2016

Different way of saying the same thing:

$\textcolor{b l u e}{y = - \frac{1}{6} x}$

#### Explanation:

$\textcolor{b l u e}{\text{General Introduction}}$

Given:$\text{ } y = 6 x$

Compare to the standard equation form of $y = m x + c$
Where $m$ is the gradient (slope)

Looking at the question we see that the gradient is 6.
This means that for 1 along on the x-axis the line has gone up 6 on the y-axis.

$\textcolor{b l u e}{y = 6 x \text{ "->" gradient "->" } m = 6}$
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$\textcolor{w h i t e}{.}$

$\textcolor{b l u e}{\text{Building the equation of the perpendicular line}}$

color(brown)("A line perpendicular to "y=mx+c" has gradient "-1/m

If $m = 6 \text{ then } - \frac{1}{m} = - \frac{1}{6}$

So the straight line perpendicular to that given has the equation

$\textcolor{b l u e}{y = - \frac{1}{m} x + c \text{ "->" } y = - \frac{1}{6} x + c}$

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$\textcolor{w h i t e}{.}$
$\textcolor{b l u e}{\text{Determine the value of the constant } c}$

This line passes through the point $\left(x , y\right) \to \left(6 , - 1\right)$

So by substitution

$y = - \frac{1}{6} x + c \text{ "->" } - 1 = - \frac{1}{\cancel{6}} \left(\cancel{6}\right) + c$

$\implies c = 0$

so the equation of the perpendicular line is:

$\textcolor{b l u e}{y = - \frac{1}{6} x}$ Jul 20, 2016

$y = - \frac{1}{6} x$

#### Explanation:

When lines are perpendicular, one slope is the negative reciprocal of the other.
In $y = 6 x$ the gradient is 6.

The slope perpendicular to this is $- \frac{1}{6}$

We now have the slope and the point $\left(6 , - 1\right)$

There is really nifty formula which applies in a case like this. We have the slope and one point and need to find the equation of the line.

$\left(y - {y}_{1}\right) = m \left(x - {x}_{1}\right)$ where the given point is $\left({x}_{1} , {y}_{1}\right)$

Substitute the values given.

$y - \left(- 1\right) = - \frac{1}{6} \left(x - 6\right) \text{ }$ multiply out and simplify.

$y + 1 = - \frac{1}{6} x + 1$

$y = - \frac{1}{6} x$