how is slope 2 and slope 1/2 parallel?

[jwk811]

how is slope 2 and slope 1/2 parallel?

if thats true can you explain it to me? i thought parallels had the same slope?

 

[jwk811]

heres the equation

y=2x+1
y=2x+2

we are supposed to find out if these two are parallel, or hit at one point etc.

i plugged in 0 to y and x separately in each and got

(0,2),(-1,0); (-1/2,0)

And put them on a graph. Couldn't tell if they were parallel but they looked it so I did
m= (y2-y1)/(x2-x1) for each and got a slope of 2 and a slope of 1/2

maybe i did something wrong...

 

[JeffM]

y=2x+1
y=2x+2

we are supposed to find out if these two are parallel, or hit at one point etc.

i plugged in 0 to y and x separately in each and got

(0,2),(-1,0); (-1/2,0)

And put them on a graph. Couldn't tell if they were parallel but they looked it so I did
m= (y2-y1)/(x2-x1) for each and got a slope of 2 and a slope of 1/2

maybe i did something wrong...

First of all, the equations above are in slope-intercept form. Two lines that are parallel have the same slope, and two lines that have the same slope are parallel. (Similarly, two lines that have slopes that multiply to minus 1 are perpendicular.)

By inspection, what is the slope of the first equation?

By inspection, what is the slope of the second equation?

Now I cannot tell where you made your mistake because you did not show a pair of points for each line, nor did you show how you calculated the slopes.

 

[jwk811]

y=2x+1
y=2x+2

from inspection now I notice the 2 stands for the slope.

 

[JeffM]

y=2x+1
y=2x+2

from inspection now I notice the 2 stands for the slope.

You should show how you calculated the slopes (even though you did not need to in this case) because otherwise we cannot explain why you went wrong.

 

[jwk811]

y=2x+1
y=2(0)+1
y=1 when x=0
0=2x+1
-1 -1
-1=2x
/2 /2
x=-1/2 when y=0
(0,1),(-1/2,0)

y=2x+2
y=2(0)+2
y=2 when x=0
0=2x+2
-2 -2
-2=2x
/2 /2
x=-1 when y=0
(0,2),(-1,0)

m=(y2-y1)/(x2-x1)
m=(0-1)/(-1/2-0)
m=-1/(-1/2)
m=1/(1/2)
m=0.5


EDIT*** 1/(1/2) = 2

Nevermind! Thanks!

 

[lookagain]

First of all, the equations above are in slope-intercept form.


Two lines that are parallel have the same slope, \(\displaystyle \ \ \ \ \)or each line has no slope.


and two lines that have the same slope are parallel. \(\displaystyle \ \ \ \ \)That is not necessarily true.*


(Similarly, two lines that have slopes that multiply to minus 1 are perpendicular.) \(\displaystyle \ \ \ \ \)**

* For example, the lines y = 2x and 3y = 6x have the same slope, but they are not parallel lines. They are the same line.


** Also, a line with a zero slope and a line with an undefined slope are perpendicular to each other.

 

[pka]

* For example, the lines y = 2x and 3y = 6x have the same slope, but they are not parallel lines. They are the same line.
** Also, a line with a zero slope and a line with an undefined slope are perpendicular to each other.

It is worth noting that any line in \(\displaystyle \Re^2 \) is parallel to itself. That is the standard definition in axiomatic geometry.

Moreover \(\displaystyle y=2x~\&~3y=6x \) are two expressions that represen one line.

Any two vertical lines are parallel.


If \(\displaystyle \ell_1~\&~\ell_2 \) are two lines with slopes \(\displaystyle m_1~\&~m_2 \) are their slopes then if \(\displaystyle {\ell _1} \bot {\ell _2}\text{ if and only if }\) one in vertical and one horizontal or else \(\displaystyle m_1\cdot m_2=-1 \).

 

[HallsofIvy]

It is worth noting that any line in \(\displaystyle \Re^2 \) is parallel to itself. That is the standard definition in axiomatic geometry.

Not in any axiomatic geometry I have ever seen! For example, Wikipedia says "In geometry, parallel lines are lines in a plane which do not meet. The assumed existence and properties of parallel lines are the basis of Euclid's parallel postulate. Two lines in a plane that do not intersect or touch at any point are said to be parallel."

"oracle" says "Parallel lines are coplanar lines that do not intersect."

Moreover \(\displaystyle y=2x~\&~3y=6x \) are two expressions that represen one line.

Any two vertical lines are parallel.


If \(\displaystyle \ell_1~\&~\ell_2 \) are two lines with slopes \(\displaystyle m_1~\&~m_2 \) are their slopes then if \(\displaystyle {\ell _1} \bot {\ell _2}\text{ if and only if }\) one in vertical and one horizontal or else \(\displaystyle m_1\cdot m_2=-1 \).

 

[pka]

Not in any axiomatic geometry I have ever seen! For example, Wikipedia says "In geometry, parallel lines are lines in a plane which do not meet. The assumed existence and properties of parallel lines are the basis of Euclid's parallel postulate. Two lines in a plane that do not intersect or touch at any point are said to be parallel." "oracle" says "Parallel lines are coplanar lines that do not intersect."

I quite agree with you that two parallel lines have no points in common.
But when speaking of a line we do not mean two it's one line.

R L Moore's student Ed Moise makes the point that the idea of parallel is that of an equivalence class.
Hence the saying "have the same slope", reflexive, symmetric, and transitive.

I was a graduate assistant for several NSF summer workshops given by one of Moore's students and based upon an old set of his notes. That is how I learned to do it that way.