# Lines in the Plane and Slope

## Introduction

The simplest mathematical model for relating two variables is the linear equation ${\displaystyle y=mx+b}$ (Slope-intercept form). This equation is called Linear because its graph is a line. ${\displaystyle m}$ is the slope and ${\displaystyle (0,b)}$ is the y-intercept.

## Finding the slope ${\displaystyle m}$

For instance, suppose you want to find the slope of the line passing through the distinct points ${\displaystyle (x_{1},x_{2})}$ and ${\displaystyle (y_{1},y_{2})}$.

 ${\displaystyle Slope={\frac {y_{2}-y_{1}}{x_{2}-x_{1}}}={\frac {y_{1}-y_{2}}{x_{1}-x_{2}}}}$


Exercises Find the slope of the line passing through the distinct points below

1) ${\displaystyle (-6,2)}$ and ${\displaystyle (3,20)}$

Solution:
${\displaystyle 2}$

2)${\displaystyle (3,-7)}$ and ${\displaystyle (-3,-7)}$

Solution:
${\displaystyle 0}$

3)${\displaystyle (3,-2)}$ and ${\displaystyle (-3,1)}$

Solution:
${\displaystyle {\frac {-1}{2}}}$

## Writing the linear equation given a slope and a point on the line

 Point-Slope Form of the Equation of a Line
The equation of the line with slope  passing through the point ${\displaystyle (x_{1},y_{1})}$ is
${\displaystyle y-y_{1}=m(x-x_{1})}$


Notice: In order to write this equation, we need a point and a slope given

Exercises Find the equation of the line line given the information below

1) slope ${\displaystyle m=3}$ and goes through ${\displaystyle (1,2)}$

Solution:
Apply the formula with ${\displaystyle m=3}$ , ${\displaystyle x_{1}=1}$ and ${\displaystyle y_{1}=2}$ to get the result
${\displaystyle y-2=3(x-1)}$

## Writing the linear equation given two points on the line

Given two point ${\displaystyle (x_{1},y_{1})}$ and ${\displaystyle (x_{2},y_{2})}$ are on the line. To find the equation of this line:

First, use the formula to find the slope

Then, apply the point-slope formula with the slope we just found and one of the given points.

Exercises Find the equation of the line passing through the distinct points below

1) ${\displaystyle (4,3)}$ and ${\displaystyle (0,-5)}$

Solution:
${\displaystyle m=slope={\frac {3-(-5)}{4-0}}={\frac {8}{4}}=2}$
Apply the point-slope formula with slope ${\displaystyle m=2}$ and the given point ${\displaystyle (4,3)}$ ( I choose ${\displaystyle (4,3)}$ in this case, but ${\displaystyle (0,-5)}$ will give the same result) to get
${\displaystyle y-3=2(x-4)}$

## Notes:

A vertical line goes through has equation of the form ${\displaystyle x=a}$ where ${\displaystyle a}$ is any constant.