Lines in the Plane and Slope

Introduction

The simplest mathematical model for relating two variables is the linear equation Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y=mx+b } (Slope-intercept form). This equation is called Linear because its graph is a line. $m$ is the slope and $(0,b)$ is the y-intercept.

Finding the slope $m$

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

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

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

1) $(-6,2)$ and $(3,20)$

Solution:
$2$

2)$(3,-7)$ and $(-3,-7)$

Solution:
$0$

3)$(3,-2)$ and $(-3,1)$

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

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 $(x_{1},y_{1})$ is 
 $y-y_{1}=m(x-x_{1})$

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 $m=3$ and goes through $(1,2)$

Solution:
Apply the formula with $m=3$ , $x_{1}=1$ and $y_{1}=2$ to get the result
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y-2=3(x-1)}

Writing the linear equation given two points on the line

Given two point Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (x_1,y_1) } and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (4,3)} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (0,-5)}

Solution:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m=slope=\frac {3-(-5)}{4-0}=\frac {8}{4}=2}
Apply the point-slope formula with slope Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m=2 } and the given point Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (4,3) } ( I choose Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (4,3) } in this case, but Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (0,-5)} will give the same result) to get
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y-3=2(x-4) }

Notes:

A vertical line goes through has equation of the form Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x=a } where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a } is any constant.

This page were made by Tri Phan