Difference between revisions of "Lines in the Plane and Slope"

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==Introduction==
 
==Introduction==
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The simplest mathematical model for relating two variables is the linear equation <math> y=mx+b </math> (Slope-intercept form). This equation is called ''Linear'' because its graph is a line. <math> m </math> is the slope and <math> (0,b) </math> is the y-intercept.
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The simplest mathematical model for relating two variables is the linear equation <math> y=mx+b </math> (Slope-intercept form). This equation is called ''Linear'' because its graph is a line. <math> m </math> is the slope and   <math> (0,b) </math> is the y-intercept.
  
 
==Finding the slope <math> m </math>==
 
==Finding the slope <math> m </math>==
 
For instance, suppose you want to find the slope of the line passing through the distinct points <math> (x_1,x_2) </math> and <math> (y_1,y_2) </math>.
 
For instance, suppose you want to find the slope of the line passing through the distinct points <math> (x_1,x_2) </math> and <math> (y_1,y_2) </math>.
  
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<div class="boxed">
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  <math>Slope =\frac {y_2-y_1}{x_2-x_1} =\frac {y_1-y_2}{x_1-x_2}</math>
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<math>Slope =\frac {y_2-y_1}{x_2-x_1} =\frac {y_1-y_2}{x_1-x_2}</math>
 
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</div>
 
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'''Exercises'''
 
'''Exercises'''
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|}
 
|}
  
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==Writing the linear equation==
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==Writing the linear equation given a slope and a point on the line==
  
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<div class="boxed">
 
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'''Point-Slope Form of the Equation of a Line'''
 
  
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The equation of the line with slope  passing through the point <math>(x_1,y_1)</math> is  
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  '''Point-Slope Form of the Equation of a Line'''
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  The equation of the line with slope  passing through the point <math>(x_1,y_1)</math> is  
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  <math>y-y_1=m(x-x_1)</math>
  
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<math>y-y_1=m(x-x_1)</math>
 
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</div>
 
  
 
''Notice:'' In order to write this equation, we need a point and a slope given
 
''Notice:'' In order to write this equation, we need a point and a slope given
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 +
'''Exercises'''
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Find the equation of the line line given the information below
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'''1)''' slope <math> m=3 </math> and goes through <math>(1,2)</math>
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{| class = "mw-collapsible mw-collapsed" style = "text-align:left;"
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!Solution: &nbsp;
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|-
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|Apply the formula with <math> m=3 </math> , <math>x_1=1</math> and <math>y_1=2</math> to get the result
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|-
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|<math style="vertical-align: -5px">y-2=3(x-1)</math>
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|-
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|}
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==Writing the linear equation given two points on the line==
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Given two point <math> (x_1,y_1) </math> and <math>(
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x_2,y_2)</math> are on the line. To find the equation of this line:
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First, use the formula to find the slope
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Then, apply the point-slope formula with the slope we just found and one of the given points.
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'''Exercises'''
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Find the equation of the line passing through the distinct points below
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'''1)''' <math>(4,3)</math> and <math>(0,-5)</math>
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{| class = "mw-collapsible mw-collapsed" style = "text-align:left;"
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!Solution: &nbsp;
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|-
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|<math style="vertical-align: -5px"> m=slope=\frac {3-(-5)}{4-0}=\frac {8}{4}=2</math>
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|-
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|Apply the point-slope formula with slope <math> m=2 </math> and the given point <math> (4,3) </math> ( I choose <math> (4,3) </math> in this case, but <math>(0,-5)</math> will give the same result) to get
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|-
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|<math style="vertical-align: -5px"> y-3=2(x-4) </math>
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|-
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|}
  
 
==Notes:==
 
==Notes:==
 
A vertical line goes through has equation of the form <math> x=a </math> where <math> a </math> is any constant.
 
A vertical line goes through has equation of the form <math> x=a </math> where <math> a </math> is any constant.
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[[Math_22| '''Return to Topics Page''']]
  
 
'''This page were made by [[Contributors|Tri Phan]]'''
 
'''This page were made by [[Contributors|Tri Phan]]'''

Latest revision as of 06:50, 19 July 2020

Introduction

The simplest mathematical model for relating two variables is the linear equation (Slope-intercept form). This equation is called Linear because its graph is a line. 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 } is the slope 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,b) } is the y-intercept.

Finding the 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 }

For instance, suppose you want to find the slope of the line passing through the distinct points 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,x_2) } 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 (y_1,y_2) } . 

 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 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) 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 (-6,2)} 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 (3,20)}

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 2}

2)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 (3,-7)} 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 (-3,-7)}

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 0}

3)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 (3,-2)} 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 (-3,1)}

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 \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 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)}
 is 
 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-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 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=3 } and goes through 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 (1,2)}

Solution:  
Apply the formula with 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=3 } , 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=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 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 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.

Return to Topics Page

This page were made by Tri Phan

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