Subject: Trigonometry

# Lines

## Trigonometry.Lines History

Hide minor edits - Show changes to output

September 29, 2010
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(:table border=3 cellpadding=3 cellspacing=0 align=center:)

(:cellnr:)

%red%'''The main equation for a line is''':%%

(:cellnr:)

%red%'''The main equation for a line is''':%%

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(:tableend:)

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(:table border=3 cellpadding=3 cellspacing=0 align=center:)

(:cellnr:)

%red%'''The slope of a line is''':%%

(:cellnr:)

%red%'''The slope of a line is''':%%

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(:tableend:)

September 29, 2010
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!! Equation of A Line

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!! The Slippery Slope Equation You Need to Know

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!! Y-Intercept

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!!! Example:

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!! Parallel Lines

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!! Intersecting Lines

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!! Perpendicular Lines

September 29, 2010
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Find the equation for a line passing through points (7,-5) and (-5, 3).

to:

Find the equation for a line passing through points (7,-5) and (-5, 3).\\

September 29, 2010
by -

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*[[#parallel | 2 ~~parallel ~~lines?]]

to:

*[[#parallel | Are 2 lines parallel?]]

September 29, 2010
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*[[#perpendicular | Perpendicular Lines]]

Changed lines 84-85 from:

Ok, let's get real deep in this line business. Now, how do you know if two lines are parallel? The slopes, ''m'', for each line will be the same. How can we figure this out?

to:

Ok, let's get real deep in this line business. Now, how do you know if two lines are parallel? The slopes, ''m'', for each line will be the same (e.g. m = 3 for both lines, or m = {$ \frac {1}{3} $} or any other constant). How can we figure this out?

Changed lines 87-90 from:

First, we would need two points on ~~one~~ ~~line ~~AB and two points on ~~the~~ ~~other~~ ~~line~~ ~~CD. ~~Let's say these points are as follows:

%center%A: (4,1) B: (10,4) C: (-7,4) D: (-9,3)

%center%A: (4,1) B: (10,4) C: (-7,4) D: (-9,3)

to:

First, we would need two points on {$ \overleftrightarrow{AB} $} and two points on {$ \overleftrightarrow{CD} $}. Let's say these points are as follows:

%center%A: (4,1) B: (10,4) & C: (-7,4) D: (-9,3)

%center%A: (4,1) B: (10,4) & C: (-7,4) D: (-9,3)

Changed lines 94-99 from:

%center%'+

%center%'+{$ m_{AB}=0.5

to:

'+{$ m_{AB}=\frac{4-1}{10-4} = \frac{3}{6} = \frac{1}{2} $}+'

Changed lines 98-103 from:

%center%'+

%center%'+{$ m_{CD}=0.5

to:

'+{$ m_{CD}=\frac{3-4}{-9-(-7)} = \frac{-1}{-2} = \frac {1}{2} $}+'

Changed lines 107-108 from:

Again we need two points on each line, and this time we use the equation of both lines and set equal to each other to find where the two lines share one point with coordinates {$ (x_{~~EFGH~~}, Y_{~~EFGH~~}) $}.

to:

Again we need two points on each line, and this time we use the equation of both lines and set equal to each other to find where the two lines share one point with coordinates {$ (x_{int}, Y_{int}) $}.

Changed lines 111-114 from:

%center% E: (-3,-6) F: (5,2) G: (-1,4) H: (2,-5)

'''Find Slopes of Both Lines:'''

'''Find Slopes of Both Lines:'''

to:

%center% E: (-3,-6) F: (5,2) & G: (-1,4) H: (2,-5)

'''1) Find Slopes of Both Lines:'''

'''1) Find Slopes of Both Lines:'''

Changed lines 117-118 from:

'''Y-Intercepts of Both Lines:'''

to:

'''2) Y-Intercepts of Both Lines:'''

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'''Plug and Chug: Set Both Line Equations Equal to One Another'''

to:

'''3) Plug and Chug: Set Both Line Equations Equal to One Another'''

Changed lines 139-152 from:

%center%~~divide~~ ~~both sides by 4 gives:~~

%center%'+{$x=1$}+'

Now, we plug x=~~2~~ into either of the two line equations:

%center%'+{$ y_{EF}=1(1)~~+(~~-3~~)~~$}+'

%center%'+{$ y_{EF}=1-3$}+'

~~%center%~~'~~+{$~~ ~~y_{EF}=-2$}+'~~

So, we can check for fun by plugging the same x-value into our other equation of the line GH:

%center%

Now, we plug x=

%center%

%center%'+{$ y_{EF}=1-3$}+'

So, we can check for fun by plugging the same x-value into our other equation of the line GH:

to:

%center%'''Answer:''' '+{$x=1$}+'

'''4) Now, we plug x=1 into either of the two line equations we found above:'''

%center%'+{$ y_{EF}= 1x - 3 = 1(1) -3$}+'

%center%'+{$ y_{EF}=1-3 = - 2 $}+'

'''Check:''' So, we can check for fun by plugging the same x-value into our other equation of the line GH (if you've found the intersection point the equations should equal each other when x = 1):

'''4) Now, we plug x=1 into either of the two line equations we found above:'''

%center%'+{$ y_{EF}= 1x - 3 = 1(1) -3$}+'

%center%'+{$ y_{EF}=1-3 = - 2 $}+'

'''Check:''' So, we can check for fun by plugging the same x-value into our other equation of the line GH (if you've found the intersection point the equations should equal each other when x = 1):

Changed lines 151-157 from:

%center%'+{$ y_{GH}=-3+1$}+'

~~%center%'+{$~~ ~~y_{GH}=-2$}+'~~

Of ~~course,~~ ~~because~~ ~~both~~ ~~lines~~ ~~intersect~~ ~~at ~~the ~~point~~ ~~(1,-2)~~ ~~by~~ ~~setting~~ ~~the ~~equations ~~equal ~~to ~~one~~ ~~another ~~(~~and~~ ~~assuming~~ ~~y~~ ~~is~~ ~~equal)~~ ~~the~~ ~~answer~~ ~~will~~ ~~provide~~ ~~the~~ ~~x-coordinate~~ ~~when~~ ~~y~~ ~~values~~ ~~are~~ ~~equal.~~ ~~You~~ ~~need~~ ~~to~~ ~~only~~ ~~take~~ ~~this~~ ~~process~~ ~~one~~ ~~step~~ ~~further~~ ~~and~~ ~~and~~ ~~solve~~ ~~for~~ ~~y,~~ ~~which~~ ~~should~~ ~~provide~~ ~~the~~ ~~same~~ ~~answer~~ ~~for~~ ~~both~~ ~~equations,~~ ~~as shown above~~.

Congratulations! You have learned about line equation, slope, y-intercept, parallel and intersecting lines.

Of

Congratulations! You have learned about line equation, slope, y-intercept, parallel and intersecting lines.

to:

%center%'+{$ y_{GH}=-3+1 = -2 $}+'

As you can see, when you set the x coordinate of both equations to 1 (which you solved for when setting both equations to equal each other), they both give you the answer of -2. Therefore''' your answer is (1, -2).'''

[[#perpendicular]]

!!! Perpendicular Lines

''Congratulations! You have learned about line equation, slope, y-intercept, parallel and intersecting lines.''

As you can see, when you set the x coordinate of both equations to 1 (which you solved for when setting both equations to equal each other), they both give you the answer of -2. Therefore''' your answer is (1, -2).'''

[[#perpendicular]]

!!! Perpendicular Lines

''Congratulations! You have learned about line equation, slope, y-intercept, parallel and intersecting lines.''

September 29, 2010
by -

Changed lines 76-78 from:

'''1) Find m:''' {$ m = \frac{3-(-5)}{-5-7} = \frac {8}{~~-~~12} = \frac {~~-~~2}{3} $}\\

'''2) Find b:''' {$ b = y-m*x = -5-\left({-2\over 3}\right * 7 = -5 - \frac{-14}{3} = \frac{~~-~~1}{3} $}\\

'''3) Put it all together:''' {$ y = mx + b = \frac {~~-~~2}{3} - \frac {1}{3} $}

'''2) Find b:''' {$ b = y-m*x = -5-\left({-2\over 3}\right * 7 = -5 - \frac{-14}{3} = \frac{

'''3) Put it all together:''' {$ y = mx + b = \frac {

to:

'''1) Find m:''' {$ m = \frac{3-(-5)}{-5-7} = -\frac {8}{12} = -\frac {2}{3} $}\\

'''2) Find b:''' {$ b = y-m*x = -5-\left({-2\over 3}\right) * 7 = -5 - \frac{-14}{3} = -\frac{1}{3} $}\\

'''3) Put it all together:''' {$ y = mx + b = -\frac {2}{3}x - \frac {1}{3} $}

'''2) Find b:''' {$ b = y-m*x = -5-\left({-2\over 3}\right) * 7 = -5 - \frac{-14}{3} = -\frac{1}{3} $}\\

'''3) Put it all together:''' {$ y = mx + b = -\frac {2}{3}x - \frac {1}{3} $}

September 29, 2010
by -

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%red%The main equation for a line is:%%

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'+{$$ \frac{rise}{run} $$}+'

to:

'+{$$ m = \frac{rise}{run} $$}+'

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%center%'+

%center%'+{$ m = 3 $}+'

to:

'+{$$ m = \frac{10-(-5)}{2-(-3)} = \frac{15}{5} = 3 $$}+'

Changed lines 59-67 from:

%center%'+

So

to:

'+{$$ m = \frac{-5-(10)}{2-(-3)} = \frac{-15}{5} = -3 $$}+'

So this is why it's important to keep your points x-values & y-values coordinated. When your slope is wrong, this will affect your calculated y-intercept, and the problems will follow you throughout any problem, so be sure to take your time and keep your ducks in a row, especially for a test! Practice organizing the information on your page in a way that you can commit to memory and perform the calculations the same each time.

So this is why it's important to keep your points x-values & y-values coordinated. When your slope is wrong, this will affect your calculated y-intercept, and the problems will follow you throughout any problem, so be sure to take your time and keep your ducks in a row, especially for a test! Practice organizing the information on your page in a way that you can commit to memory and perform the calculations the same each time.

Changed lines 65-73 from:

The term y-intercept is a fancy way of saying the point where the line crosses the y-axis, or, where {~~$ ~~x~~=0~~ ~~$}~~ in the equation of a line. We can rearrange the equation of a line to understand for any point on the line, b=y-mx. Be sure to use the same point for both x and y values when plugging and chugging. We can find the y-intercept of a line PQ with a slope m=2/5. Let's use point P (10, -5) for our example and plug these values into the equation below.

~~%center%~~'+{$ b=-5-(2~~/~~5)*10 $}+'

~~%center%'+{$~~ ~~b=-5-4~~ $}~~+'~~

%center%'+{$ ~~b=-9 ~~$}~~+'~~

So ~~we~~ ~~know~~ ~~for~~ ~~line~~ ~~PQ~~ ~~with~~ ~~slope~~ ~~m=2/5~~ ~~that ~~the ~~y-intercept~~ ~~occurs at the point ~~(0,-9).

%center%'+{$

So

to:

The term y-intercept is a fancy way of saying the point where the line crosses the y-axis, or, where {x = 0} in the equation of a line. We can rearrange the equation of a line to understand for any point on the line, '''b=y-mx'''. Be sure to use the same point for both x and y values when plugging and chugging. We can find the y-intercept of a line PQ with a slope m=2/5. Let's use point P (10, -5) for our example and plug these values into the equation below.

'+{$$ b= y-mx = -5-\left({2\over 5}\right)*10 = -5-4 = -9 $$}+'

So we know for {$ \overleftrightarrow{PQ} $} with slope m=2/5 that the y-intercept occurs at the point (0,-9).

!!!! Example:

Find the equation for a line passing through points (7,-5) and (-5, 3).

(:toggle id=box1 show='Show Answers' init=hide button=1:)

>>id=box1 border='1px solid #999' padding=5px bgcolor=#edf<<

'''1) Find m:''' {$ m = \frac{3-(-5)}{-5-7} = \frac {8}{-12} = \frac {-2}{3} $}\\

'''2) Find b:''' {$ b = y-m*x = -5-\left({-2\over 3}\right * 7 = -5 - \frac{-14}{3} = \frac{-1}{3} $}\\

'''3) Put it all together:''' {$ y = mx + b = \frac {-2}{3} - \frac {1}{3} $}

>><<

'+{$$ b= y-mx = -5-\left({2\over 5}\right)*10 = -5-4 = -9 $$}+'

So we know for {$ \overleftrightarrow{PQ} $} with slope m=2/5 that the y-intercept occurs at the point (0,-9).

!!!! Example:

Find the equation for a line passing through points (7,-5) and (-5, 3).

(:toggle id=box1 show='Show Answers' init=hide button=1:)

>>id=box1 border='1px solid #999' padding=5px bgcolor=#edf<<

'''1) Find m:''' {$ m = \frac{3-(-5)}{-5-7} = \frac {8}{-12} = \frac {-2}{3} $}\\

'''2) Find b:''' {$ b = y-m*x = -5-\left({-2\over 3}\right * 7 = -5 - \frac{-14}{3} = \frac{-1}{3} $}\\

'''3) Put it all together:''' {$ y = mx + b = \frac {-2}{3} - \frac {1}{3} $}

>><<

September 29, 2010
by -

Changed lines 5-7 from:

to:

(:table border=1 cellpadding=5 cellspacing=0:)

(:cellnr bgcolor=#d4d7ba colspan=14 align=center:) '+'''The Breakdown'''+'

(:cellnr:)

*[[#equation | The equation of a line]]

*[[#slope | How to find the slope of a line]]

*[[#yintercept | How to find the y-intercept of a line]]

*[[#parallel | 2 parallel lines?]]

*[[#intersecting | Intersecting lines (& where?)]]

(:tableend:)

When you write about lines in verbiage, you should name a line by the points along that line. See below, this line would be called line AB, or {$ \overleftrightarrow{AB} $}. Later on, we will talk about how these points really are the defining features of the line.

(:cellnr bgcolor=#d4d7ba colspan=14 align=center:) '+'''The Breakdown'''+'

(:cellnr:)

*[[#equation | The equation of a line]]

*[[#slope | How to find the slope of a line]]

*[[#yintercept | How to find the y-intercept of a line]]

*[[#parallel | 2 parallel lines?]]

*[[#intersecting | Intersecting lines (& where?)]]

(:tableend:)

When you write about lines in verbiage, you should name a line by the points along that line. See below, this line would be called line AB, or {$ \overleftrightarrow{AB} $}. Later on, we will talk about how these points really are the defining features of the line.

Changed lines 19-21 from:

So let's move on and talk about what your teachers want you to know about lines. If you have two lines, it's important to know if they are intersecting or parallel lines in math. I think we all know what those two terms mean, but how can you prove one or the other? Well, ~~a~~ ~~line's~~ ~~equation~~ ~~can~~ ~~pretty~~ ~~much~~ ~~tell~~ ~~you~~ ~~everything~~ ~~you'd~~ ~~want~~ ~~to~~ ~~know~~ ~~about~~ ~~a line~~.

to:

So let's move on and talk about what your teachers want you to know about lines. If you have two lines, it's important to know if they are intersecting or parallel lines in math. I think we all know what those two terms mean, but how can you prove one or the other? Well, it all starts with the equation of a line, then we'll add on from there.

[[#equation]]

[[#equation]]

Changed lines 24-28 from:

%center%'+{$

to:

How can you tell what direction a line is traveling? Or where it intersects the x and y axis? Or how steep it is? To start, every line has its own equation to fully define it and set it apart from all other lines. All you need to find the equation of any line is either:

* The slope of the line, ''m,'' as well as one point along the line (call it ''x,y'')

* Or two points on a line (call it x'_1_',y'_1_' & x'_2_',y'_2_')

*A line either parallel or perpendicular to the line and one additional point.

'''The main equation for a line is:'''

'+{$$ y = mx + b $$}+'

''where:''

->'''m''' = is the slope of the line\\

'''b''' = the y-intercept of the line\\

'''x & y''' = Given any x coordinate you can find the corresponding y coordinate and vice versa.

[[#slope]]

* The slope of the line, ''m,'' as well as one point along the line (call it ''x,y'')

* Or two points on a line (call it x'_1_',y'_1_' & x'_2_',y'_2_')

*A line either parallel or perpendicular to the line and one additional point.

'''The main equation for a line is:'''

'+{$$ y = mx + b $$}+'

''where:''

->'''m''' = is the slope of the line\\

'''b''' = the y-intercept of the line\\

'''x & y''' = Given any x coordinate you can find the corresponding y coordinate and vice versa.

[[#slope]]

Changed lines 43-46 from:

We remember in the ~~cartesian~~ ~~coordinate~~ system, there are two axes, ''x'' and ''y'' which represent horizontal and vertical movement on a 2D graph. You'll need at least two points to define a line, because there are an infinite number of lines that could possibly be running through one point. Let's call those two points (''x'''_1_', ''y'''_1_') and (''x'''_2_', ''y'''_2_'). The equation to define the slope of a line is as follows:

~~%center%~~'+{$ m = \frac{y_{2}-y_{1}}{x_{2}-x_{1}} $}+'

to:

We remember in the [[CoordinateGeometry | cartesian coordinate system]], there are two axes, ''x'' and ''y'' which represent horizontal and vertical movement on a 2D graph. You'll need at least two points to define a line, because there are an infinite number of lines that could possibly be running through one point. Let's call those two points (''x'''_1_', ''y'''_1_') and (''x'''_2_', ''y'''_2_'). The equation to define the slope of a line is as follows:

'+{$$ m = \frac{y_{2}-y_{1}}{x_{2}-x_{1}} $$}+'

'+{$$ m = \frac{y_{2}-y_{1}}{x_{2}-x_{1}} $$}+'

Changed lines 49-50 from:

to:

'+{$$ \frac{rise}{run} $$}+'

Changed line 73 from:

to:

[[#yintercept]]

Added line 85:

[[#parallel]]

Added line 113:

[[#intersecting]]

September 06, 2010
by -

Changed lines 147-149 from:

Of course, because both lines intersect at the point (1,-2) by setting the equations equal to one another (and assuming y is equal) the answer will provide the x-coordinate when y values are equal. You need to only take this process one step further and and solve for y, which should provide the same answer for both equations.

to:

Of course, because both lines intersect at the point (1,-2) by setting the equations equal to one another (and assuming y is equal) the answer will provide the x-coordinate when y values are equal. You need to only take this process one step further and and solve for y, which should provide the same answer for both equations, as shown above.

Congratulations! You have learned about line equation, slope, y-intercept, parallel and intersecting lines.

Congratulations! You have learned about line equation, slope, y-intercept, parallel and intersecting lines.

September 06, 2010
by -

Changed lines 99-100 from:

%center% E: (-3,-6) F: (5,2) G: (-1,4) H: (2,-5)~~$}~~

to:

%center% E: (-3,-6) F: (5,2) G: (-1,4) H: (2,-5)

Added line 102:

Added line 106:

Changed lines 110-112 from:

Now, plug these into an equation form for the two lines~~. ~~:

to:

Now, plug these into an equation form for the two lines:

Added line 124:

September 06, 2010
by -

Changed lines 10-12 from:

So let's move on and talk about what your teachers want you to know about lines. If you have two lines, it's important to know if they are intersecting or parallel lines in math. I think we all know what those two terms mean, but how can you prove one or the other? Well, a line's ~~slope~~ can pretty much tell you everything you'd want to know about a line.

to:

So let's move on and talk about what your teachers want you to know about lines. If you have two lines, it's important to know if they are intersecting or parallel lines in math. I think we all know what those two terms mean, but how can you prove one or the other? Well, a line's equation can pretty much tell you everything you'd want to know about a line.

Changed line 15 from:

Now, all lines that have the same slope are parallel and will never intersect, so how do you tell them apart mathematically? Each line has its own equation to fully define it and set it apart from all other lines. This equation is based on the slope ''m'' ~~that~~ ~~we~~ ~~just~~ ~~defined~~ ~~as~~ ~~well~~ ~~as~~ ~~coordinates of a point on the line ~~(''x,y'') and ''b'', the y-intercept.

to:

Now, all lines that have the same slope are parallel and will never intersect, so how do you tell them apart mathematically? Each line has its own equation to fully define it and set it apart from all other lines. This equation is based on the slope ''m'' as well as coordinates of a point (''x,y'') on the line and ''b'', the y-intercept.

September 06, 2010
by -

Added lines 13-19:

!!! Equation of A Line

Now, all lines that have the same slope are parallel and will never intersect, so how do you tell them apart mathematically? Each line has its own equation to fully define it and set it apart from all other lines. This equation is based on the slope ''m'' that we just defined as well as coordinates of a point on the line (''x,y'') and ''b'', the y-intercept.

%center%'+{$ y = mx + b $}+'

Now, all lines that have the same slope are parallel and will never intersect, so how do you tell them apart mathematically? Each line has its own equation to fully define it and set it apart from all other lines. This equation is based on the slope ''m'' that we just defined as well as coordinates of a point on the line (''x,y'') and ''b'', the y-intercept.

%center%'+{$ y = mx + b $}+'

Deleted lines 50-55:

!!! Equation of A Line

Now, all lines that have the same slope are parallel and will never intersect, so how do you tell them apart mathematically? Each line has its own equation to fully define it and set it apart from all other lines. This equation is based on the slope ''m'' that we just defined as well as coordinates of a point on the line (''x,y'') and ''b'', the y-intercept.

%center%'+{$ y = mx + b $}+'

September 06, 2010
by -

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Added line 36:

Added line 66:

'''Points on Both Lines:'''

Added line 71:

'''Slopes of both lines:'''

Added line 93:

'''Points on Both Lines:'''

Added line 100:

'''Find Slopes of Both Lines:'''

Added line 103:

'''Y-Intercepts of Both Lines:'''

Added line 106:

'''Plug and Chug: Set Both Line Equations Equal to One Another'''

Changed lines 110-111 from:

We will first solve for x by setting the two equations equal to one another:

to:

We will first solve for x by assuming ''y'' is equal and setting the two equations equal to one another:

Added line 113:

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September 06, 2010
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Added lines 23-41:

One thing you must be sure of is that the order of subtraction between the two points in the numerator and denominator stays the same. It doesn't matter which point goes first, so long as you reference the same point first on top and bottom (numerator and denominator).

For example if you were to take the y-value of point G first in the numerator and then you accidentally used the x-value for H first in the denominator, the solution could possibly give you a negative slope from the correct answer. Let's use two points G: (-3, -5) and H: (2, 10) for an example:

'''Correct Calculation:'''

%center%'+{$ m = \frac{10-(-5)}{2-(-3)} $}+'

%center%'+{$ m = \frac{15}{5} $}+'

%center%'+{$ m = 3 $}+'

'''Incorrect Calculation:'''

%center%'+{$ m = \frac{-5-(10)}{2-(-3)} $}+'

%center%'+{$ m = \frac{-15}{5} $}+'

%center%'+{$ m = -3 $}+'

So this is why it's important to keep your points, x-values, y-values and all coordinated information straight and organized. When your slope is wrong, this will affect your calculated y-intercept, and the problems will follow you throughout a problem, so be sure to take your time and keep your ducks in a row, especially for a test! Practice organizing the information on your page in a way that you can commit to memory and perform the calculations the same each time.

For example if you were to take the y-value of point G first in the numerator and then you accidentally used the x-value for H first in the denominator, the solution could possibly give you a negative slope from the correct answer. Let's use two points G: (-3, -5) and H: (2, 10) for an example:

'''Correct Calculation:'''

%center%'+{$ m = \frac{10-(-5)}{2-(-3)} $}+'

%center%'+{$ m = \frac{15}{5} $}+'

%center%'+{$ m = 3 $}+'

'''Incorrect Calculation:'''

%center%'+{$ m = \frac{-5-(10)}{2-(-3)} $}+'

%center%'+{$ m = \frac{-15}{5} $}+'

%center%'+{$ m = -3 $}+'

So this is why it's important to keep your points, x-values, y-values and all coordinated information straight and organized. When your slope is wrong, this will affect your calculated y-intercept, and the problems will follow you throughout a problem, so be sure to take your time and keep your ducks in a row, especially for a test! Practice organizing the information on your page in a way that you can commit to memory and perform the calculations the same each time.

September 06, 2010
by -

Changed line 26 from:

Now, all lines that have the same slope are parallel and will never intersect, so how do you tell them apart mathematically? Each line has its own equation to fully define it and set it apart from all other lines. This equation is based on the slope ''m'' that we just defined as well as coordinates of a point on the line and ''b'', the y-intercept.

to:

Now, all lines that have the same slope are parallel and will never intersect, so how do you tell them apart mathematically? Each line has its own equation to fully define it and set it apart from all other lines. This equation is based on the slope ''m'' that we just defined as well as coordinates of a point on the line (''x,y'') and ''b'', the y-intercept.

September 06, 2010
by -

Added line 35:

Added line 37:

Changed line 74 from:

%center%~~'+{$~~ E(-3,-6) F(5,2) G(-1,4) H(2,-5)$}~~+'~~

to:

%center% E: (-3,-6) F: (5,2) G: (-1,4) H: (2,-5)$}

September 06, 2010
by -

Changed lines 3-4 from:

There are hundreds of definitions of a line in figurative speech, but if you want to pass your math test, it will help to remember that a line is straight and never-ending. When you draw a line on paper, you should show that the line never ends by using arrows on either end pointing far, far into the distance.

to:

There are hundreds of definitions of a line in figurative speech, but if you want to pass your math test, it will help to remember that a line is straight and never-ending. Your teacher may try to trick you on your test and ask you how long the ''line'' is, but remember you can only measure a ''segment'' between two points. Lines have infinite length. When you draw a line on paper, you should show that the line never ends by using arrows on either end pointing far, far into the distance.

Changed lines 15-16 from:

We remember in the cartesian coordinate system, there are two axes, ''x'' and ''y'' which represent horizontal and vertical movement on a 2D graph. You'll need two points to define a line, because there are an infinite number of lines that could possibly be running through one point. Let's call those two points (''x'''_1_', ''y'''_1_') and (''x'''_2_', ''y'''_2_'). The equation to define the slope of a line is as follows:

to:

We remember in the cartesian coordinate system, there are two axes, ''x'' and ''y'' which represent horizontal and vertical movement on a 2D graph. You'll need at least two points to define a line, because there are an infinite number of lines that could possibly be running through one point. Let's call those two points (''x'''_1_', ''y'''_1_') and (''x'''_2_', ''y'''_2_'). The equation to define the slope of a line is as follows:

Changed lines 26-27 from:

Now, all lines that have the same slope are parallel and will never intersect, so how do you tell them apart mathematically? Each line has its own equation to fully define it and set it apart from all other lines. This equation is based on the slope ''m'' that we just defined as well as coordinates of a point on the line and ''b'', the y-intercept. ~~The term y-intercept is a fancy way of saying the point where the line crosses the y-axis.~~

to:

Now, all lines that have the same slope are parallel and will never intersect, so how do you tell them apart mathematically? Each line has its own equation to fully define it and set it apart from all other lines. This equation is based on the slope ''m'' that we just defined as well as coordinates of a point on the line and ''b'', the y-intercept.

Changed lines 32-37 from:

The y-intercept ~~occurs~~ ~~where ~~a ~~line~~ ~~crosses~~ ~~the~~ ~~y-axis,~~ ~~or,~~ where ~~{$~~ ~~x=0~~ ~~$}~~ ~~in ~~the ~~equation~~ ~~of~~ ~~a~~ ~~line.~~ ~~We~~ ~~can~~ ~~rearrange~~ the equation of a line ~~to~~ ~~understand~~ ~~for~~ ~~any~~ ~~point~~ ~~on~~ ~~the~~ line~~,~~ ~~b=y-mx.~~ ~~Be~~ ~~sure~~ ~~to~~ ~~use~~ the ~~same~~ ~~point~~ ~~for~~ ~~both~~ ~~x~~ ~~and~~ ~~y~~ ~~values~~ ~~when~~ ~~plugging~~ ~~and~~ ~~chugging.~~ ~~We~~ ~~can~~ ~~find~~ ~~the~~ ~~y-intercept~~ ~~once~~ ~~we~~ ~~know~~ ~~the~~ ~~slope~~ ~~of~~ ~~a~~ ~~line~~ PQ~~.~~ ~~Let's~~ ~~use~~ ~~these~~ ~~points~~ ~~for~~ ~~our~~ ~~example: ~~point P

%center%'+{$ ~~b=-6-~~(~~1~~)~~*-3~~ ~~$}+'~~

%center%'+{$ ~~b=-6+3~~ $}

%center%'+{$

%center%'+{$

to:

The term y-intercept is a fancy way of saying the point where the line crosses the y-axis, or, where {$ x=0 $} in the equation of a line. We can rearrange the equation of a line to understand for any point on the line, b=y-mx. Be sure to use the same point for both x and y values when plugging and chugging. We can find the y-intercept of a line PQ with a slope m=2/5. Let's use point P (10, -5) for our example and plug these values into the equation below.

%center%'+{$ b=-5-(2/5)*10 $}+'

%center%'+{$ b=-5-4 $}+'

%center%'+{$ b=-9 $}+'

So we know for line PQ with slope m=2/5 that the y-intercept occurs at the point (0,-9).

%center%'+{$ b=-5-(2/5)*10 $}+'

%center%'+{$ b=-5-4 $}+'

%center%'+{$ b=-9 $}+'

So we know for line PQ with slope m=2/5 that the y-intercept occurs at the point (0,-9).

Changed lines 76-84 from:

We also need to fine the y-intercept, ''b'', where a line cross the y-axis, and x equals zero. ~~See~~ ~~above~~ ~~for~~ ~~the~~ ~~full~~ ~~explanation,~~ ~~but~~ ~~we'll~~ ~~go~~ ~~quickly~~ ~~here~~ ~~for~~ ~~the~~ ~~sake~~ of ~~moving~~ ~~quickly.~~ ~~For~~ ~~any~~ point ~~on the line~~,~~ b=y~~-~~mx.~~ ~~Be~~ ~~sure~~ ~~to~~ ~~use~~ ~~the~~ ~~same~~ ~~point~~ ~~for~~ ~~both~~ ~~x~~ ~~and~~ ~~y~~ ~~values~~ ~~when~~ ~~plugging~~ ~~and~~ ~~chugging.~~ ~~We~~ ~~can~~ ~~find~~ ~~the~~ y~~-intercept~~ ~~because~~ ~~we~~ ~~already~~ ~~know~~ ~~the~~ ~~slope~~ ~~of~~ ~~each~~ ~~line.~~ ~~Let's~~ ~~try~~ ~~line~~ ~~EF~~ ~~first,~~ ~~using~~ ~~point~~ ~~E~~ ~~for~~ ~~our~~ ~~values.~~

%center%'+{$ ~~b=~~-~~6-(1~~)~~*-3~~ $}+'

%center%'+{$~~b=-6~~+~~3~~ $}

Now, ~~plug~~ ~~these~~ ~~into~~ ~~an~~ ~~equation~~ ~~form~~ ~~for~~ ~~the two lines:~~

%center%'+{$ ~~y_{EF}=~~ ~~1x+b~~ $} ~~and~~ ~~{$~~ ~~y_{GH}~~ ~~=~~ ~~-3x~~+~~b ~~$}+'

%center%'+{$

Now,

to:

We also need to fine the y-intercept, ''b'', where a line cross the y-axis, and x equals zero. Again, check yourself on your own scratch paper with the correct answer. The y-intercept of line EF occurs at the point (0,-3) and the y-intercept of line GH occurs at the point (0,1)

Now, plug these into an equation form for the two lines. :

%center%'+{$ y_{EF}= 1x+(-3) $} and {$ y_{GH} = -3x+1 $}+'

We will first solve for x by setting the two equations equal to one another:

%center%'+{$ 1x+(-3) = -3x+1 $}+'

%center%'+{$ +3x = +3x $}+'

%center%'+{$ 4x-3 = 1 $}+'

%center%'+{$ +3 = +3 $}+'

%center%'+{$ 4x = 4 $}+'

%center%divide both sides by 4 gives:

%center%'+{$x=1$}+'

Now, we plug x=2 into either of the two line equations:

%center%'+{$ y_{EF}=1(1)+(-3)$}+'

%center%'+{$ y_{EF}=1-3$}+'

%center%'+{$ y_{EF}=-2$}+'

So, we can check for fun by plugging the same x-value into our other equation of the line GH:

%center%'+{$ y_{GH}=-3(1)+1$}+'

%center%'+{$ y_{GH}=-3+1$}+'

%center%'+{$ y_{GH}=-2$}+'

Of course, because both lines intersect at the point (1,-2) by setting the equations equal to one another (and assuming y is equal) the answer will provide the x-coordinate when y values are equal. You need to only take this process one step further and and solve for y, which should provide the same answer for both equations.

Now, plug these into an equation form for the two lines. :

%center%'+{$ y_{EF}= 1x+(-3) $} and {$ y_{GH} = -3x+1 $}+'

We will first solve for x by setting the two equations equal to one another:

%center%'+{$ 1x+(-3) = -3x+1 $}+'

%center%'+{$ +3x = +3x $}+'

%center%'+{$ 4x-3 = 1 $}+'

%center%'+{$ +3 = +3 $}+'

%center%'+{$ 4x = 4 $}+'

%center%divide both sides by 4 gives:

%center%'+{$x=1$}+'

Now, we plug x=2 into either of the two line equations:

%center%'+{$ y_{EF}=1(1)+(-3)$}+'

%center%'+{$ y_{EF}=1-3$}+'

%center%'+{$ y_{EF}=-2$}+'

So, we can check for fun by plugging the same x-value into our other equation of the line GH:

%center%'+{$ y_{GH}=-3(1)+1$}+'

%center%'+{$ y_{GH}=-3+1$}+'

%center%'+{$ y_{GH}=-2$}+'

Of course, because both lines intersect at the point (1,-2) by setting the equations equal to one another (and assuming y is equal) the answer will provide the x-coordinate when y values are equal. You need to only take this process one step further and and solve for y, which should provide the same answer for both equations.

September 06, 2010
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Deleted line 23:

Changed lines 30-32 from:

to:

!!! Y-Intercept

The y-intercept occurs where a line crosses the y-axis, or, where {$ x=0 $} in the equation of a line. We can rearrange the equation of a line to understand for any point on the line, b=y-mx. Be sure to use the same point for both x and y values when plugging and chugging. We can find the y-intercept once we know the slope of a line PQ. Let's use these points for our example: point P

%center%'+{$ b=-6-(1)*-3 $}+'

%center%'+{$ b=-6+3 $}

Changed lines 72-81 from:

In order to write the equation of a line, {$ y=mx+b $}, you must first understand the slope ''m''.

to:

In order to write the equation of a line, {$ y=mx+b $}, you must first understand the slope ''m''. You should do this on your own scratch piece of paper, but I've discovered the two slopes {$ m_{EF} = 1 $} and {$ m_{GH} = -3 $}.

We also need to fine the y-intercept, ''b'', where a line cross the y-axis, and x equals zero. See above for the full explanation, but we'll go quickly here for the sake of moving quickly. For any point on the line, b=y-mx. Be sure to use the same point for both x and y values when plugging and chugging. We can find the y-intercept because we already know the slope of each line. Let's try line EF first, using point E for our values.

%center%'+{$ b=-6-(1)*-3 $}+'

%center%'+{$ b=-6+3 $}

Now, plug these into an equation form for the two lines:

%center%'+{$ y_{EF}= 1x+b $} and {$ y_{GH} = -3x+b $}+'

We also need to fine the y-intercept, ''b'', where a line cross the y-axis, and x equals zero. See above for the full explanation, but we'll go quickly here for the sake of moving quickly. For any point on the line, b=y-mx. Be sure to use the same point for both x and y values when plugging and chugging. We can find the y-intercept because we already know the slope of each line. Let's try line EF first, using point E for our values.

%center%'+{$ b=-6-(1)*-3 $}+'

%center%'+{$ b=-6+3 $}

Now, plug these into an equation form for the two lines:

%center%'+{$ y_{EF}= 1x+b $} and {$ y_{GH} = -3x+b $}+'

September 06, 2010
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Added lines 57-68:

It's safe to say these lines are parallel! Because '+{$ m_{AB}=m_{CD} $}+' we know their slopes are the same, and these lines will never intersect.

!!! Intersecting Lines

So, we understand the concept of parallel lines, which means we know that when two lines ''do not'' have the same slope, they will intersect. Most often your teachers will ask you to go a step further and understand where these two lines will intersect. We can identify that location as a point in the 2D cartesian coordinate graph.

Again we need two points on each line, and this time we use the equation of both lines and set equal to each other to find where the two lines share one point with coordinates {$ (x_{EFGH}, Y_{EFGH}) $}.

Our two lines will be line EF and line GH with points as follows:

%center%'+{$ E(-3,-6) F(5,2) G(-1,4) H(2,-5)$}+'

In order to write the equation of a line, {$ y=mx+b $}, you must first understand the slope ''m''.

!!! Intersecting Lines

So, we understand the concept of parallel lines, which means we know that when two lines ''do not'' have the same slope, they will intersect. Most often your teachers will ask you to go a step further and understand where these two lines will intersect. We can identify that location as a point in the 2D cartesian coordinate graph.

Again we need two points on each line, and this time we use the equation of both lines and set equal to each other to find where the two lines share one point with coordinates {$ (x_{EFGH}, Y_{EFGH}) $}.

Our two lines will be line EF and line GH with points as follows:

%center%'+{$ E(-3,-6) F(5,2) G(-1,4) H(2,-5)$}+'

In order to write the equation of a line, {$ y=mx+b $}, you must first understand the slope ''m''.

September 06, 2010
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Changed lines 41-47 from:

Now let's plug these points into the equations we looked at before to find out the slope:

%center%'+~~m_~~{AB}=\frac{4-1}{10-4}+'

%center%'+~~m_~~{AB}=\frac{3}{6}+'

%center%'+~~m_~~{AB}=0.5+'

%center%'+

%center%'+

%center%'+

to:

Now let's plug these points into the equations we looked at before to find out the slope of the line AB:

%center%'+{$ m_{AB}=\frac{4-1}{10-4} $}+'

%center%'+{$ m_{AB}=\frac{3}{6} $}+'

%center%'+{$ m_{AB}=0.5 $}+'

Now let's compare to find out the slope of the line CD:

%center%'+{$ m_{CD}=\frac{3-4}{-9-(-7)} $}+'

%center%'+{$ m_{CD}=\frac{-1}{-2} $}+'

%center%'+{$ m_{CD}=0.5 $}+'

%center%'+{$ m_{AB}=\frac{4-1}{10-4} $}+'

%center%'+{$ m_{AB}=\frac{3}{6} $}+'

%center%'+{$ m_{AB}=0.5 $}+'

Now let's compare to find out the slope of the line CD:

%center%'+{$ m_{CD}=\frac{3-4}{-9-(-7)} $}+'

%center%'+{$ m_{CD}=\frac{-1}{-2} $}+'

%center%'+{$ m_{CD}=0.5 $}+'

September 06, 2010
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Changed lines 38-49 from:

%center%~~ ~~A: (4,1) B: (10,4) C: (-7,4) D: (-9,3)

to:

%center%A: (4,1) B: (10,4) C: (-7,4) D: (-9,3)

Now let's plug these points into the equations we looked at before to find out the slope:

%center%'+m_{AB}=\frac{4-1}{10-4}+'

%center%'+m_{AB}=\frac{3}{6}+'

%center%'+m_{AB}=0.5+'

Now let's plug these points into the equations we looked at before to find out the slope:

%center%'+m_{AB}=\frac{4-1}{10-4}+'

%center%'+m_{AB}=\frac{3}{6}+'

%center%'+m_{AB}=0.5+'

September 03, 2010
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Changed line 38 from:

%center%~~{$~~ A: (4,1) B: (10,4) C: (-7,4) D: (-9,3) ~~$}~~

to:

%center% A: (4,1) B: (10,4) C: (-7,4) D: (-9,3)

September 03, 2010
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Changed line 38 from:

%center%{$ A: (4,1) B: (10,4) C: (-7,4) D: (-9,3)~~ ~~ $}

to:

%center%{$ A: (4,1) B: (10,4) C: (-7,4) D: (-9,3) $}

September 03, 2010
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Changed lines 38-42 from:

%center%{$ A: (4,1) B: (10,4) C: (-7,4) D: (-9,3) $}

Unfortunately

Unfortunately

to:

%center%{$ A: (4,1) B: (10,4) C: (-7,4) D: (-9,3) $}

September 03, 2010
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Added lines 33-38:

!!! Parallel Lines

Ok, let's get real deep in this line business. Now, how do you know if two lines are parallel? The slopes, ''m'', for each line will be the same. How can we figure this out?

First, we would need two points on one line AB and two points on the other line CD. Let's say these points are as follows:

%center%{$ A: (4,1) B: (10,4) C: (-7,4) D: (-9,3) $}

September 03, 2010
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Changed lines 5-7 from:

When you write about lines in verbiage, you should name a line by the points along that line. See below, this line would be called line AB, or the letters AB with a line on top (see the image for an example).

to:

When you write about lines in verbiage, you should name a line by the points along that line. See below, this line would be called line AB, or the letters AB with a line on top (see the image for an example). Later on, we will talk about how these points really are the defining features of the line.

Changed lines 12-15 from:

We

to:

!!! The Slippery Slope Equation You Need to Know

We remember in the cartesian coordinate system, there are two axes, ''x'' and ''y'' which represent horizontal and vertical movement on a 2D graph. You'll need two points to define a line, because there are an infinite number of lines that could possibly be running through one point. Let's call those two points (''x'''_1_', ''y'''_1_') and (''x'''_2_', ''y'''_2_'). The equation to define the slope of a line is as follows:

Added lines 23-28:

!!! Equation of A Line

Now, all lines that have the same slope are parallel and will never intersect, so how do you tell them apart mathematically? Each line has its own equation to fully define it and set it apart from all other lines. This equation is based on the slope ''m'' that we just defined as well as coordinates of a point on the line and ''b'', the y-intercept. The term y-intercept is a fancy way of saying the point where the line crosses the y-axis.

Changed lines 31-36 from:

Unfortunately

to:

The real beauty is, if you have two points, you can always use the slope equation to find out the equation of the line intersecting those two points.

Unfortunately

Unfortunately

September 03, 2010
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Changed lines 14-15 from:

We remember in the cartesian coordinate system, there are two axes, ''x'' and ''y'' which represent horizontal movement and vertical movement on a 2D graph. ~~The~~ ~~equation~~ ~~to~~ ~~define~~ ~~the~~ ~~slope~~ ~~of ~~a ~~graph~~ ~~is~~ ~~as~~ follows:

to:

We remember in the cartesian coordinate system, there are two axes, ''x'' and ''y'' which represent horizontal movement and vertical movement on a 2D graph. You'll need two points to define a line, because there are an infinite number of lines that could possibly be running through one point. Let's call those two points (''x'''_1_', ''y'''_1_') and (''x'''_2_', ''y'''_2_'). The equation to define the slope of a graph is as follows:

%center%'+{$ m = \frac{y_{2}-y_{1}}{x_{2}-x_{1}} $}+'

%center%or, most often referred to as:

%center%'+{$ \frac{rise}{run} $}+'

%center%'+{$ m = \frac{y_{2}-y_{1}}{x_{2}-x_{1}} $}+'

%center%or, most often referred to as:

%center%'+{$ \frac{rise}{run} $}+'

Changed lines 26-29 from:

Let's look at an example and see what we can find out.

to:

Unfortunately

September 03, 2010
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Changed line 20 from:

%center%'+{$ \frac{rise}{run} $}+'

to:

%center%'++{$ \frac{rise}{run} $}++'

September 03, 2010
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Changed lines 16-17 from:

{$ y = mx + b $}

to:

%center%'+{$ y = mx + b $}+'

Changed line 20 from:

{$ \frac{rise}{run} $}

to:

%center%'+{$ \frac{rise}{run} $}+'

September 03, 2010
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Changed lines 8-23 from:

%cframe width=275px border='2px solid black' padding=5px% %width=265px%Attach:LineAB.~~jpg~~

to:

%cframe width=275px border='2px solid black' padding=5px% %width=265px%Attach:LineAB.jpg

So let's move on and talk about what your teachers want you to know about lines. If you have two lines, it's important to know if they are intersecting or parallel lines in math. I think we all know what those two terms mean, but how can you prove one or the other? Well, a line's slope can pretty much tell you everything you'd want to know about a line.

If two lines have the same slope, they will never touch. All other lines will intersect at one point. Let's learn about the slope equation; this one will stay with you the rest of your mathematical life, so don't forget it, or commit it to your short-term memory bank by accident.

We remember in the cartesian coordinate system, there are two axes, ''x'' and ''y'' which represent horizontal movement and vertical movement on a 2D graph. The equation to define the slope of a graph is as follows:

{$ y = mx + b $}

Where ''x'' and ''y'' can be defined by the coordinates of a point on the line, and ''b'' represents the value at which the line crosses the y-axes, otherwise known as the y-intercept. Then ''m'' represents the multiplier for the slope of the line, typically represented by the equation as follows:

{$ \frac{rise}{run} $}

Let's look at an example and see what we can find out.

So let's move on and talk about what your teachers want you to know about lines. If you have two lines, it's important to know if they are intersecting or parallel lines in math. I think we all know what those two terms mean, but how can you prove one or the other? Well, a line's slope can pretty much tell you everything you'd want to know about a line.

If two lines have the same slope, they will never touch. All other lines will intersect at one point. Let's learn about the slope equation; this one will stay with you the rest of your mathematical life, so don't forget it, or commit it to your short-term memory bank by accident.

We remember in the cartesian coordinate system, there are two axes, ''x'' and ''y'' which represent horizontal movement and vertical movement on a 2D graph. The equation to define the slope of a graph is as follows:

{$ y = mx + b $}

Where ''x'' and ''y'' can be defined by the coordinates of a point on the line, and ''b'' represents the value at which the line crosses the y-axes, otherwise known as the y-intercept. Then ''m'' represents the multiplier for the slope of the line, typically represented by the equation as follows:

{$ \frac{rise}{run} $}

Let's look at an example and see what we can find out.

September 02, 2010
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Changed line 8 from:

%cframe width=275px border='2px solid black' padding=5px% %width=265px%Attach:LineAB.~~jpg\\~~

to:

%cframe width=275px border='2px solid black' padding=5px% %width=265px%Attach:LineAB.jpg

September 02, 2010
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Changed line 8 from:

%cframe~~%~~ ~~%~~width=~~265px~~%Attach:LineAB.~~jpg~~

to:

%cframe width=275px border='2px solid black' padding=5px% %width=265px%Attach:LineAB.jpg\\

September 02, 2010
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Added lines 1-8:

!! I Know What a Line is, But What is a Line?

There are hundreds of definitions of a line in figurative speech, but if you want to pass your math test, it will help to remember that a line is straight and never-ending. When you draw a line on paper, you should show that the line never ends by using arrows on either end pointing far, far into the distance.

When you write about lines in verbiage, you should name a line by the points along that line. See below, this line would be called line AB, or the letters AB with a line on top (see the image for an example).