# How do you write an equation of a line in function notation given the line goes through (2,3); perpendicular to 6x-7y=6?

Jun 30, 2015

Background for slopes:
$\textcolor{w h i t e}{\text{XXXX}}$The slope of a line is defined as
$\textcolor{w h i t e}{\text{XXXX}}$$\textcolor{w h i t e}{\text{XXXX}}$$\frac{\Delta y}{\Delta x}$
$\textcolor{w h i t e}{\text{XXXX}}$That is, given two points $\left({x}_{1} , {y}_{1}\right)$ and $\left({x}_{2} , {y}_{2}\right)$ on the line
$\textcolor{w h i t e}{\text{XXXX}}$the slope is
$\textcolor{w h i t e}{\text{XXXX}}$$\textcolor{w h i t e}{\text{XXXX}}$$m = \frac{{y}_{2} - {y}_{1}}{{x}_{2} - {x}_{1}}$

$\textcolor{w h i t e}{\text{XXXX}}$For a straight line the slope is the same for all pairs of points on the line
$\textcolor{w h i t e}{\text{XXXX}}$Therefore, given two fixed points (as above) and a variable point $\left(x , y\right)$ on the line
$\textcolor{w h i t e}{\text{XXXX}}$$\textcolor{w h i t e}{\text{XXXX}}$$\frac{y - {y}_{1}}{x - {x}_{1}} = \frac{{y}_{2} - {y}_{1}}{{x}_{2} - {x}_{1}}$
$\textcolor{w h i t e}{\text{XXXX}}$This can be rewritten:
$\textcolor{w h i t e}{\text{XXXX}}$$\textcolor{w h i t e}{\text{XXXX}}$$y = m \left(x - {x}_{1}\right) + {y}_{1}$

$\textcolor{w h i t e}{\text{XXXX}}$If a line has a slope of $\hat{m}$ then all lines perpendicular to it have a slope of $\frac{1}{\hat{m}}$

Slope of $6 x - 7 y = 6$
$\textcolor{w h i t e}{\text{XXXX}}$This equation can be rewritten as
$\textcolor{w h i t e}{\text{XXXX}}$$\textcolor{w h i t e}{\text{XXXX}}$$y = \left(\frac{6}{7}\right) x + \left(\frac{6}{7}\right)$
$\textcolor{w h i t e}{\text{XXXX}}$and therefore has a slope of $\left(\frac{6}{7}\right)$
$\textcolor{w h i t e}{\text{XXXX}}$Any line perpendicular to it has a slope of $\left(- \frac{7}{6}\right)$

Equation of a line through $\left(2 , 3\right)$ perpendicular to $6 x - 7 y = 6$
$\textcolor{w h i t e}{\text{XXXX}}$Using the previous discussion:
$\textcolor{w h i t e}{\text{XXXX}}$$\textcolor{w h i t e}{\text{XXXX}}$$y = \left(- \frac{7}{6}\right) \left(x - 2\right) + 3$
$\textcolor{w h i t e}{\text{XXXX}}$or, simplified and re-written in function notation
$\textcolor{w h i t e}{\text{XXXX}}$$\textcolor{w h i t e}{\text{XXXX}}$$f \left(x\right) = - \frac{7}{6} x + \frac{2}{3}$