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Slope Simplified! Learn Rise Over Run + The Slope Formula
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Mar 14, 2025
Struggling with slope? This video breaks it down step by step! Learn how to master rise over run and the slope formula with clear explanations and examples. Perfect for prealgebra students looking to understand these key concepts. Don’t forget to like, comment, and subscribe for more math tips and lessons!" Chapters:
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0:00
all right so we're talking about slopes
0:03
today right that's the reason why I had
0:05
you guys do the research on the rate of
0:09
change because that's going to lead us
0:11
now to the slope all right we're going
0:13
to talk about the slope of a line slope
0:15
of a slope of um a line and we're going
0:19
to talk about slopes in general okay so
0:21
by
0:22
definition um what is the slope by
0:25
definition the slope is uh the ratio the
0:30
ratio or the division between the rise
0:33
and the run
0:37
right the slope like I said again is the
0:40
ratio of the rise over the
0:43
run you
0:46
guys watch okay rise over the r now what
0:50
do we do with the slope the slope tells
0:52
you about the stiffness the stiffness of
0:55
a line okay how this line is steep so
0:58
the slope is going to tell you how the
1:00
line is so that's what you use a slope
1:02
for right A lot of people use slopes
1:04
when they are skating I mentioned that
1:07
all right if you're flying a kite you
1:09
can see the angle at which your kite is
1:11
flying at you can do a lot of that using
1:13
the slope okay so in this particular
1:16
case it's easy to find the slope when I
1:18
give you the rise and the Run that's
1:20
easy to do right so to find the slope
1:23
again is what we call the vertical
1:26
change over the horizontal change right
1:31
rise overrun so if I was to find the
1:34
slope of this um thing here this figure
1:38
that would be 15 over 18 right and I I
1:43
have to simplify that so that is
1:46
basically uh I divide this by three so 5
1:50
over uh six right so the slope here be
1:53
56 again this is your slope and to find
1:56
you do rise over the Run yeah so the SL
2:00
is that line going this is the slope
2:02
yeah it gives you this is the no this is
2:04
it tells you the stiffness our stiff
2:06
this is right this is your slope so this
2:08
is your slope there so I want to know
2:10
what it is in terms of value right so to
2:12
find it I'm going to do the rise over
2:15
the run and that's going to give me the
2:16
value of my slope I want to see how
2:18
steep that is right so if I want to ride
2:20
my skape I want to see can I take this
2:22
slope I don't know this is too stick for
2:24
me I'm not going to take a chance you
2:26
need to know what a slope is I mean you
2:28
can visualize it right you can tell see
2:30
wow this is too step I'm not taking this
2:32
right but you can also find the actual
2:34
value of your slope by doing that now
2:37
the next thing we going to talk about is
2:40
finding the slope of of a line that
2:43
passes through two points right how do
2:45
you find the slope of a line that passes
2:47
through through two point if you do not
2:49
have a figure like this if I give you a
2:52
line and I put it on the set of axis and
2:55
I want you to find the slope of that
2:56
line how would you do it right so this
3:00
is going to follow you all the way to
3:02
calculus 3 it's going to follow you
3:05
everywhere right once you know this
3:06
formula here is going to follow you
3:09
everywhere next year inra one you're
3:10
going to see this formula uh in NRA two
3:13
we did we did a lot of that this year
3:15
you're going to learn how to find the
3:16
equation of a line you going to learn
3:18
how to find the slope of a line that is
3:20
always going to follow you everywhere if
3:21
you doing calculus we find it too yes
3:24
what the little one after the X and Y
3:27
mean are those oh those are just subp
3:30
right so x sub One X sub2 so basically
3:33
what I want I want to differentiate
3:35
between these two point I could have
3:36
called them X2 X3 X4 X5 X5 and the y5
3:41
I'm just trying to differentiate them
3:42
they're not the same right so I will
3:45
explain when you look at the figure
3:46
right so if you see this line here
3:49
right like she said before the line is
3:53
made up of an infinite number of points
3:56
but to find the slope of a line you only
3:58
need but what two points I don't need
4:00
more than two points to find the slope
4:01
of a line okay I only need two points so
4:04
what I'm doing here is this this is my
4:07
line and I want to identify the slope
4:10
right I want to know what the slope is
4:11
so basically I want to know what the
4:13
rise over the run is so to find it we
4:16
use this formula that's going to follow
4:18
us like I said everywhere going forward
4:21
we're going to be doing this a lot we're
4:23
going to learn how to build the equation
4:25
of a line right given two point we going
4:28
to learn how to find the equation of
4:29
that line
4:30
this is interesting stuff because you
4:32
use data right if you have a set of data
4:34
let me give let me just give you an
4:36
example here okay so if you have this
4:40
here right and you have this data points
4:43
like you have a bunch of data points
4:44
here
4:47
right you have this here and you want to
4:49
study for example this let's say this is
4:52
price right and then
4:56
uh time right let's say you bought
4:59
bought a shoe right you bought a shoe
5:02
and you want to know you're a seller you
5:04
want to resell that shoe right he has
5:07
some shoes that Jordan people buy those
5:10
they sell them okay and you want to know
5:13
this is this is uh the price of the shoe
5:16
over time the price is going to change
5:18
it's changing what's happening here what
5:19
do you see about the price of the shoe
5:21
as time passes it's going up right so
5:24
let's say you bought the shoe and you
5:26
want to find an equation of this shoe
5:29
that's going to help you to figure out
5:30
how much that shoe is going to be worth
5:33
in 10 and 20 years let's say you do it
5:36
and you come up with an equation like
5:37
this Y is = to
5:39
2x + 100 right so $100 is how much you
5:44
pay for the shoe right now but every
5:46
let's assume that every month or every
5:49
year the shoes going by $2 in 10 years
5:52
how much that Sho is going to be worth
5:53
how would you figure it out you're going
5:55
to replace this by what 10 and find it
5:58
right so this is why we use a linear
6:00
equation if it shows you a linear model
6:02
then you can use an equation like this
6:04
to figure out so if I'm a seller I'm
6:07
going to be like all right do I really
6:09
want to wait 10 years to make $20 on a
6:10
shoe is that something you want to do
6:12
maybe yes maybe not do I want to wait
6:15
five no now if he's like growing up at
6:17
$2 every month that's even better
6:20
because in 10 months now that shes wor
6:22
120 in 10 years is going to be worth a
6:25
lot more money than that right so this
6:27
is why you use this equation especially
6:29
a linear model linear model means that
6:32
it shows you the pattern of a line does
6:35
that make sense right so this is why we
6:39
use this kind of thing so don't say this
6:40
is useless it's not at all it's very
6:43
useful okay so now now we're going to
6:46
learn to to you too because you get who
6:49
just said to me who just said I Heard a
6:51
Voice I don't know yeah can you drink
6:55
your water yeah you don't have to ask me
6:57
that you can drink it all right all
6:59
right so now let me give you an example
7:02
right so let's say we have this line
7:03
here we're going to try to find the
7:05
slope of several lines so let's assume
7:08
that this is uh 1 2 3 4 5 and this is 1
7:12
2 3 4 so this is 54 right and another
7:16
point is
7:18
here one two this is
7:20
two and this is let's say two all right
7:24
so this is 22 now how do I find the
7:28
slope of this line
7:30
using this two point we're going to
7:31
apply the formula that we just learned
7:34
right we're going to apply this
7:36
formula goal is to identify X1 y1 and
7:39
then X2 Y2 and it's very simple stuff so
7:42
all you have to do is just put a label
7:44
on top of it I'm going to label this X1
7:47
I'm going to label this X2 X uh y1 right
7:51
and I'm going to label this X2 and Y2
7:54
okay that's all I'm doing here and then
7:56
I can use this to find the slope the
7:59
rest is just plugging the stuff in there
8:01
it's not rocket science it's really
8:02
simple stuff right so the slope by
8:06
definition is called M ask me why they
8:09
don't use the letter s instead of M I
8:11
don't know I don't know why they use M
8:14
instead of M I don't know why they did
8:16
that but everywhere you go in Algebra 2
8:19
algebra one the slope is always going to
8:20
be used they always going to use a
8:22
letter N for the slope okay so to find
8:25
the slope now I know Y2 I know X2 I know
8:29
y1 and X1 I'm just going to plug them in
8:31
right so it's going to be Y2 - y1 which
8:35
is what 4 right - 2 because that's Y 2 -
8:41
y1 / X2 5 - 2 5 - 2 because X1 is 2
8:50
right so therefore the slope m is going
8:52
to be 2 over 3 2/3 okay the slope is 2/3
8:57
now be careful how you do some people
9:00
when you apply this formula sometimes
9:01
they put the X on top and the Y is on
9:03
the bottom you have to watch out for
9:05
that
9:05
yes doesn't matter which one you label
9:08
X1 and y1 it doesn't matter which one
9:10
youel X1 or y1 right I can have I could
9:13
have called this X1 y1 X2 Y2 it will
9:16
give me absolutely same answer good
9:18
question it does not matter but what
9:20
matters is you keep the the right order
9:23
right if I put Y2 here I cannot put here
9:26
X1 it has to be Y2 Y2 Y2 X2 and then y1
9:30
X1 you have to make sure you keep that
9:32
order but if you don't you're going to
9:34
get um you may get a negative answer
9:37
right now the slope is 2/3 it's a
9:40
positive slope that means the line is
9:42
doing what the line is going up right
9:45
when the slope is positive that mean you
9:47
have a line that is going up now this is
9:49
telling you again the stiffness of this
9:51
line now let's do some other examples
9:54
let's say and I want you all to try out
9:56
on your own right so let's say we have
9:58
this here
10:01
let's say we have a line that looks like
10:03
this right let's assume that this is uh
10:11
-1 six right in
10:15
here this is three and one I want you to
10:20
find the slope right again I'm going to
10:22
use this two point I want you go ahead
10:24
and find the slope and tell me what
10:26
you're going to do and tell me
10:29
do you think the slope is going to be
10:30
negative or
10:32
positive for this one here negative
10:35
negative why is it negative this that's
10:38
not why
10:39
it's it's going down that means if you
10:42
find a positive slope that means you did
10:44
what you did something wrong right so
10:46
you need to make sure that you always
10:48
check the direction of the line if the
10:50
line is going down the slope is
10:53
certainly negative so therefore if you
10:55
calculate and you find a positive slope
10:57
you got to tell yourself something is
10:59
right okay so to do that again we going
11:01
to label this as
11:03
X1 y1 X2 Y 2 and the rest is just
11:09
plugging these numbers in our formula so
11:11
go ahead and find it and tell me what
11:13
you got
11:29
youall get
11:36
it so you drawing stuff while I'm like
11:40
asking you to do work and you also
11:42
drawing oh that's fantastic all
11:47
right so let me know what y'all got who
11:49
did
11:51
it yes what you got how' you do it set
11:54
it up for me I did n = 1 - 6
12:00
3 -1 all right 1 - 6 over 3 -1 I change
12:06
to like
12:08
double 1 - 6 3 + 1 all right
12:14
= 5 over4 right so your slope is 54 if
12:18
you don't have this then something is
12:20
wrong somewhere right everybody got the
12:22
54 uh good all right so now let me let
12:26
me make it a little more complicated
12:28
right now let me let me ask you to find
12:29
the
12:33
slope of a
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line that
12:39
passes
12:43
through uh
12:47
-25
12:49
and 0 and -7 I want you to find the
12:53
slope of the line that passes through
12:55
these two points so how you going to do
12:57
it
12:59
I want you to find the slope of line now
13:01
I didn't give you any picture right you
13:03
don't really need a picture to find the
13:04
slope do you need a picture we only need
13:07
the formula right now we just need to
13:09
label our points and then go ahead and
13:12
find this okay so I want you to find the
13:16
slope of the line that passes through
13:18
-25 and then 07 yes go
13:22
ahead make sure in the morning you go
13:25
empty your bladder before you come in
13:26
here you be go this is like aing thing
13:28
for you every day I see you going to the
13:30
bathroom so make sure you trying to like
13:32
do that in the morning she disr to
13:46
you got it almost almost all
13:51
right so then I'm going to label these
13:54
ones
14:00
who wants to help me with it all right
14:03
go ahead first TN
14:08
yeah
14:11
2- huh
14:13
I2 so how' you get2
14:16
-5 how's that possible what's your X1 y1
14:20
did you put a label on them you see what
14:22
I did here
14:25
right so all right let's see what you
14:27
got so I
14:30
=
14:33
-- all right over 0 - -2 excellent over
14:38
0 - -2 good so I changed that into -7 +
14:43
5 over 0 + 2 all right and that got me
14:48
-2 over 2 all right so then I divided
14:52
that by two M and I got zero huh I got
14:55
zero you got what you about two you get
14:57
what
14:59
not zero it's not -2 / 2 is 1
15:05
right1 everybody got
15:07
one who didn't get the negative
15:10
one I did everything right but then I I
15:17
make all right so you see what happens
15:20
here right so basically we have to make
15:23
sure we keep our signs right again
15:26
everything that we doing now is coming
15:27
together from the last
15:31
um the first quarter all the way now we
15:34
we are pretty much using the same Tech
15:36
the all the technique that we've used
15:38
okay so now let me ask you to find
15:39
another slope here I have two slopes
15:41
that I want you to find and tell me what
15:43
do you
15:44
notice so let's say we have this line
15:47
here right are we going to take a test
15:49
before the quarter
15:51
Ends don't
15:53
that so let's have this line here right
15:56
it's not our choice
15:59
all right so we have
16:05
one all right so this is
16:10
uh this point is uh three and three here
16:14
and this is -2 and three I want you to
16:17
find the slope right and I also want you
16:20
to find the slope of the next line let
16:22
me draw another line here
16:34
so we have another line we have two
16:35
separate
16:49
lines go sit
16:56
down one
17:01
[Music]
17:04
all right so do this one first and then
17:07
do this one
17:09
next and I want somebody El to tell me
17:12
EV I want you to tell me what you got
17:13
for the first one here and then uh what
17:16
you got did you do it yet I'm not all
17:19
right let tell me when you're
17:22
finish
17:25
and me see
17:29
K uh kiny I want you to tell me about
17:32
that one here tell me how you set it up
17:34
and then find
17:45
it we have to put the label on it right
17:48
we're going to call this
17:50
X1 y1 we're going to call this X2 Y2
17:54
we're going to do the same to here
17:56
X1 y1
17:59
and then X2 Y2 and all we're going to do
18:01
is just apply the formula yes so people
18:06
use this
18:09
for they do whatever yeah you can use it
18:12
for whatever you got your options all
18:16
right every are you done with the first
18:19
one so what you
18:23
got = 3 - okay over uh
18:30
3- excellent so what did I give
18:33
you uh so I I added
18:39
the that's five
18:42
right is Zer so what's 0 over five Z
18:47
zero right so what do you notice about
18:49
any horizontal line what is the slope of
18:52
any line that is horizontal is always
18:54
going to be what zero no matter what
18:57
it's always going to
18:59
it's flat so there's no slope is there
19:01
any thickness none so it's always going
19:03
to be zero right so any horizontal line
19:06
is always going to have a slope of zero
19:08
you don't even have to calculate it you
19:10
just look at it and then make that you
19:12
can make that call now the next one we
19:16
have something here
19:18
right
19:20
so how do we do it
19:26
again so what would that be then
19:29
oh yeah -2 - what
19:33
1 over 2 - 2 right now look at that it's
19:38
-3 over 0 what is wrong with this
19:41
picture can you divide a number by Zero
19:43
no what you get to calculus it's going
19:45
to be Infinity right we that's going to
19:48
give us Infinity what does that say the
19:51
line is vertical it's so step there's
19:54
not even like it's just boom so there's
19:56
no slope on it so it would
19:59
undef right when you get it's undefine
20:02
because you can't have a number over
20:04
zero no it's
20:08
undefine because you cannot have a
20:10
number over zero you can do zero over a
20:12
number but you can't have a number over
20:15
zero just going to leave
20:18
it it be Infinity but we don't do that
20:21
because we not there we still right so
20:24
we're not doing that yet we just going
20:25
to say it's on the F right I still get
20:28
this
20:34
wait tell me right any number over Z is
20:38
on the you can't have that for
20:41
the2 isn't that a dou you change it to
20:45
no no no it's -2 - one it's NO2 -- I
20:49
have two and
20:51
I so now what I want you to do is uh
20:54
turn a break I want you to go ahead and
20:57
uh
21:00
anything hey you run your mouth more
21:02
than you write that's
21:08
enough all right so now we going to I'm
21:10
going to give you a couple of examples
21:12
here that I want you to work on right so
21:28
a
21:29
kite a kite yes a kite thought say a
21:35
kite is
21:47
[Music]
21:59
and
22:00
a I just
22:04
Rec
22:06
stoping I can't see
22:14
it what
22:16
cursive I can't read
22:19
cursive I can't read the birthday card
22:22
my grandpa gives me all right so
22:25
basically the guy John is flying a kite
22:28
right he's fing the kite and the kite
22:31
now listen to this we're going to try
22:32
and draw this picture right he's fing
22:34
the kite and the kite is a horizontal
22:36
distance of 20 ft from John's position
22:39
right so let's assume that the kite is
22:40
here right this is the kite right and
22:44
then this is the string here John is
22:46
right here right he's holding the kite
22:49
he's holding it his arm and then he's
22:52
flying the Kite and pretty much he's a
22:55
distance they telling us that the kite
22:57
right is this a horizontal distance of
22:59
20 ft from John's position
23:03
so let's let's watch this right this is
23:05
it so if the car is a horizontal basis
23:08
of 20 ft from John's
23:11
position
23:13
who
23:16
really that's it no more thank
23:20
you so 20 ft from here right that's the
23:23
horizontal distance and then the
23:26
vertical distance is 70 ft right
23:29
so now the question is they want you to
23:32
find the slope of the kite so how would
23:36
you do
23:39
it rise overun rise over run right here
23:42
we have a picture we don't have to use
23:44
the formula it's just going to to rise
23:46
or
23:49
run rise what's the
23:51
r right so the slope will
23:54
be 70 over 20 which is basically
23:59
right which is 3.5 so this is but the
24:02
thing is is you have to be able to draw
24:04
the picture of what's going on right how
24:07
would you be able to draw that picture
24:08
to know what's going on here you need to
24:10
be able to draw that picture right so
24:12
the guy is flying the Kite and then is
24:14
saying that the kite is a horizontal
24:16
distance of 20 ft from John's position
24:18
so 20 ft so that kite is 20 ft from his
24:22
position and then the vertical distance
24:24
of 70 ft right and I find the slope of
24:28
the string this is the string
24:32
right so we need to find the slope of
24:35
the string he's holding it so therefore
24:37
this is how you do it all right all
24:41
right so now what I want to do is I'm
24:42
going to give you the some work to do
24:45
now I don't