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Understanding linear functions, How To identify and write them in the standard form
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Feb 19, 2025
In our last section, we learned how to analyze relations and functions. In this section, we'll learn how to identify linear relations and functions, as well as learn how to write linear equations in the standard form. Chapters 00:00 Introduction 01:02 What is a linear relation? 02:02 Examples of linear and non linear functions 04:00 What is a linear function? 05:34 Examples of linear functions 09:02 How to write a linear function in the standard form 19:02 How to use the x and y intercepts to graph a linear function
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0:01
all right so we're going to start uh
0:02
section 2.2 which is which is linear
0:06
relations and functions all right so if
0:09
you look at my board here you're going
0:11
to see that I drew a function okay this
0:13
is linear all right it's a line it
0:16
depicts a line and the equation of this
0:18
function is x + y = 6 right so in
0:22
general in general
0:25
relations that form a straight line are
0:28
called linear Rel
0:30
relations let me say that again
0:33
relations we talked about relations
0:35
remember we talk about what a relation
0:36
is we we buildt and we have a diagram we
0:39
build relations and we talk about when a
0:41
relation is a function so we discussed
0:43
all of that right you guys know when a
0:45
function is onto when the function is
0:47
one one all that type of stuff we talked
0:49
about last week and now we're going to
0:51
talk about specific functions which are
0:54
called linear relations or linear
0:57
functions okay so now linear the word
1:00
line occurs in there so linear means it
1:02
forms a line if you look at this point
1:05
this dot here dot dot dot dot if you
1:07
draw a line across them it's linear it's
1:10
not crooked it has to be a straight line
1:13
right it's a straight line so these are
1:14
called a linear function now if you have
1:17
a set of dot that do not form a line is
1:20
called nonlinear nonlinear relation
1:23
right say if I have this here if I were
1:26
to graph this and if I have this that
1:30
this this and that if I was to correct
1:32
the dot you see if I was to correct it
1:35
this is this is not linear it doesn't
1:38
form a line it form a curve or this we
1:41
talked about the par right so that will
1:43
not be linear that would be nonlinear
1:45
because it's not a straight line okay so
1:48
that's nonlinear so we're going to talk
1:50
about linear today now if you look here
1:53
I laid out several examples of linear
1:57
equations and nonlinear now if you look
2:00
on the first one we're going to talk
2:01
about in a minute these are called This
2:03
is the standard form okay standard form
2:06
of a linear equation we're going to
2:08
discuss it in a minute but these are
2:09
just examples of linear equations if you
2:12
were to graph this it will give you a
2:14
line if you were to graph this it will
2:16
give you a line as well and then this is
2:18
also a line which is a horizontal line
2:22
now here what is the difference I have
2:24
2X + 6 Y2 = -25 yes a square anytime a
2:31
square occurs you do not have a line
2:34
you're going to have some form of curve
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right you're going to have a curve and
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here what do I have here that shows that
2:40
this is not a linear equation square
2:42
root so square root is also very
2:44
interesting because it looks like a
2:45
curve and square root of x usually goes
2:48
like this right so that's not a linear
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equation and now here what do you see
2:52
that shows that this is not a linear yes
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I have not only one variable I have two
2:57
variables and then they're connected x *
2:59
y right so that would be nonlinear I
3:02
have x + x * y = 8 that's not a linear
3:08
equation okay we're going to next see
3:10
how we can tell now here what do I have
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here one/ x x is in the denominator so
3:20
therefore this is not linear if you were
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to graph this it would give you what a
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hyperbola have you heard about a
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hyperbola not yet so it looks like this
3:30
something like this uh this is called
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hyperbola right so you're going to see
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that it's going to form some form of
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curve like that we're going to talk
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about maybe either albra 2 or if you do
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preal you're going to talk about
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hyperbolous so these are nonlinear
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equations right now we're going to come
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up with a general formula for all linear
3:49
equations that's going to facilitate the
3:51
task of identifying there okay so now a
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linear function what is it by definition
3:59
by definition what is a linear function
4:01
a linear function is a function with
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ordered pairs okay ordered pairs it's
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not nonordered is ordered pair yes what
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is the difference between a nonlinear
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equation and a linear equation nonlinear
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yeah non Lineum is not a line yeah the
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equations oh here yeah oh if you look
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here right you see you have 6 y s
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because of the square if you were to
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draw this you would get a curve okay you
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won't get a line and here you have a
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square root so square root shows that
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this is not a line so if it has like
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weird things in it yes but we going I'm
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going to show you exactly how to tell so
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that you don't have to even guess all
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right okay so now here so a linear
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function by definition is a function
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with ordered pairs that satisfy a linear
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equation linear right a linear function
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this is the key they can all be written
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in the form what F ofx = mx + b anytime
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you can establish function that looks
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like this you automatically know it is a
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linear equation to answer your question
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right so here for example can you write
5:09
this in this format no way Jose it right
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it ain't happening so we can't we we
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can't have that so it's not going to
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happen you have to be able to put in the
5:20
form MX plus b for that to like work if
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it doesn't work therefore it's not a
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linear equation okay now example let's
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do some examples here real quick I have
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F ofx = 8 - 3/4 * X so I'm going to call
5:41
on you then to see if you can do it um
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shade do you think this is a linear
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equation yes or no yes yes because
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what's your slope m is M is-3 over 4 and
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then B is eight in other way you can
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rewrite this right if you were to say
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you know what I don't like the way this
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looks I want to write this so it looks
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just like a linear equation I can
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rewrite this as -3
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4x + 8 does anybody have a problem this
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do you all get it yes right so because I
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can write this in the form mx + b where
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m is 34 and then B is 8 therefore this
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is a linear
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equation right and you also have your
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ti3 so if you let's say you struggling
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just graph it to see if you have that
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par right if forms the of a
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linear line then it is a linear equation
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what about this one
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2X is it linear no why CU 2x 2X and we
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can't write this in the form MX plus b
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so therefore no it's not a linear what
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about this one here 3x y - 4 no no why
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because the X and Y are the X and Y are
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here right now multiply the two VAR of
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being multiply to one another it's not
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later on we find out that this is
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interesting because this is a equation
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with two unknown here we can graph this
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at some point what about that f ofx = x
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+ 5 is this linear no nonlinear because
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this is no because you cannot be you
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cannot write this in the form MX plus b
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and you have a radical right a radical
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or square root so therefore is not
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linear not linear now what about that F
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ofx = 3 is it linear yes why because
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it's three
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three right so how can you put this in
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the form MX plus b so that way make
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sense yes um 0x+ 0x + 3 right good good
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answer 0x + 3 so yes this is linear
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because the slope is what zero right you
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can still right this form MX plus P you
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have a zero slope so therefore this is
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still linear right does that make sense
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all right this is
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linear now what about the last one um
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kayy what you think is this linear no
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why not um
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because I don't know but I know it's it
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is
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linear yes is it or left so why so how
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can you rewrite this Kay try how would
8:17
you write this in the form it's linear
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what are you even saying how can you
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write this in the form MX plus b so that
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way you can prove that this is a linear
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equation
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I don't know think I am thinking all
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right well Telly you got an answer for
8:40
[Music]
8:44
me
8:48
mhm
8:51
MH zero right that's it so yes this is
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linear right yes because the slope is
8:58
what2 and the Y intercept of B is zero
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so therefore yes this is linear right so
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this chapter should not be too hard
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right so now we're going to learn one
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thing we're going to learn how to write
9:10
the equation in the standard form how do
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we go from the F ofx = mx + b to what we
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call the standard form this is the
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standard form of the of any linear
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equation right so with this form there
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are several things that we need to know
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it's always written in the form a x + b
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y = c and a has to be more than or equal
9:34
to Zer and a and b are not both zero
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okay they cannot both be zero either B
9:40
is zero or a is z they can't be at the
9:43
same time zero that's just not going to
9:45
happen one of them can be but the other
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one cannot like they cannot both be zero
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at the same time does that make sense so
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this is what I'm trying to say here ax +
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b y is = C where a is greater than or
9:58
equal to Zer and A and B are not both
10:02
zero okay so now I have this equation
10:06
here 3 10 x = 8 y - 15 and I'm going to
10:12
try to put this in a standard form right
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so here's the thing when you want to put
10:17
something okay do you need this yes but
10:19
it's okay I'll keep it can I it's okay
10:22
yeah you can res that I raas this one I
10:24
I'll leave that one there okay no it's
10:27
okay you can erase it now I'm fine you
10:29
sure yeah
10:32
okay
10:34
Mary what you calling Mary for I'm just
10:36
saying H
10:38
you all right so we're going to try to
10:40
put this in the standard form right now
10:42
here's the thing a cannot be a fraction
10:45
B cannot be a fraction and C cannot be a
10:48
fraction right they have to be integers
10:51
does that make sense they have to be
10:53
integers now this question we have -3
11:01
over 10
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x equal to 8 y - 15 what is my first
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step I'm trying to go from here to here
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what is my first step yes sir subtract 8
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y subtract 8 y all right I'm going to
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subtract 8 y so now I
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have -2 over
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10x - 8 y = -15 now what's the very next
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step after that
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mhm 10 multiply by 10 to do what to get
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rid of the fraction to get rid of the
11:39
fraction right because we have to make
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sure they are what integers yeah right
11:44
they have to be integers so I have to
11:46
multiply this by 10 does that make sense
11:48
sh you get it can you get you get it uh
11:52
I was writing all right so we are trying
11:54
to go from here right to here that make
11:59
sense and we don't we want a b and c to
12:02
be integers we don't want them to be
12:04
fractions so the first thing we did we
12:06
subtract the a y and now we multiply
12:08
everything by 10 so that way we can get
12:10
rid of the denominator all right so I'm
12:13
going to times this by 10 this by 10 and
12:16
this by 10 so if I do this I
12:20
get uh 3x right because the 10 cancel
12:24
out - 80 y = -150 I'm still done because
12:30
A and
12:31
B have to be what they have to be
12:34
positive integers they can't be negative
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right a can be a a has to be more than
12:38
or equal to zero but they cannot both be
12:40
zero at the same time it doesn't matter
12:42
if B is negative a must be positive yes
12:46
sir you just multip by multiply by1
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right I have to multiply everything by
12:52
netive 1 so I can get rid of the
12:54
negative right or if you feel like
12:56
you're too fast you don't need to go
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through this step you can multiply
12:59
everything by what -10 you could have
13:01
done that here you know you could have
13:03
done that or if you want to just be mo
13:06
you multiply by netive 1 here at the end
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right so now I get
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3x + 8 y = 15 80 y 80 y thank you equals
13:18
to 150 right you times everything by it
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by yes if you do one thing for one time
13:24
you have to do the same thing for every
13:25
time okay you can't just do for one if
13:27
you change one you have to change
13:29
everything else here okay okay all right
13:33
now let's try to do this for this one
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here we got 2
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y =
13:39
4x + 5 so what would be my first step
13:43
subtract 4X subtract
13:45
4X um you know what that's a good idea
13:48
but you make us do more work right I'd
13:51
rather bring the 2 y here and then move
13:53
the five here that's fine let's just do
13:55
that subtract it four that's fine that's
13:57
a good idea too
14:00
yeah -4x + 2 y = 5 then
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what get rid negative by doing
14:10
what by negative one right that was
14:13
actually better than what I was saying
14:15
so that's better than my step would have
14:17
been even longer than that so this is
14:19
actually better right multiply by1 now
14:22
we got
14:23
4x - 2 Y
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= and that's it all right you guys can
14:29
try to do this on your own 3x - 6 y = -
14:32
9 = 0 wouldn't it be
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POS no I'm MP by negative one all five
14:38
negative yeah yeah right as long as a is
14:41
positive we're good yeah I saw your like
14:44
five is negative yeah all right so how
14:49
about I got yeah
14:59
hard that last one's hard which one the
15:01
last one yeah which one 3x - 6
15:05
y -9 all right let's try that one 3x - 6
15:10
y
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3x - 6 y - 9 = 0 so what we do
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first add 9
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right so that is
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3x 6 y = 9 is there anything El to do no
15:27
we're done you
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joking yeah I was like yeah
15:36
man all right so with that said I want
15:39
us to work on the world problem and then
15:41
we can start working on um so it says
15:45
that the growth rate of a sample let me
15:47
let me write that on the board real
15:51
quick where's my do look so good why'
15:54
you your look good
16:06
dude look so
16:17
good maybe help me the
16:22
test you use different on one
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I can see a large variety of it's okay
16:32
if you
16:53
are wait so the X and the Y should be on
16:56
the same side yeah yeah that's right in
16:59
standard form mhm put it
17:18
for did
17:27
went so look at this so they tell us
17:30
that the growth rate of a sample
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of grass is given by this function right
17:38
and now the question is how tall is the
17:41
SLE after 3 days right X stands for the
17:44
number of days since the initial uh
17:46
thing that happened right so how tall is
17:48
the grass after 3 days what do you do
17:51
how do you
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find three and it's fine right yeah and
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then now what if I replace X by zero
17:57
what would I tell you zero now what
18:00
would I if I if I want to find F of Z
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right is 5.9 * what 0 plus 325 right
18:08
which is 3.25 how would you interpret
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this that is the uh grph now the grass
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initial height of the G right grass now
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which is the initial all right just want
18:21
to make sure you guys get right now yeah
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right now all right can you just go to
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the so now we're going to also talk
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about Y intercept and then that closes
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this section x and y intercept this is
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really super easy right so if I have an
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equation let me give you an equation and
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we going to talk about the X and the Y
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intercept and then are we're done we're
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done with this lecture and then that
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means we begin to do uh work
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start so we have
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2x - 3 y + 8 is = z so how do we use the
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inter the X and Y intercept to graphic
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function we going to learn how to do
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that right graph this
19:07
function graph the function
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using
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DX and y intercept
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right who knows what the x intercept is
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and
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Y I have a question M on the homework
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for number 6 through 11 it says Identify
19:30
a b and c but there's no which one I
19:34
didn't give you a homework yet yeah you
19:36
did you did it's on the board oh well
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yeah I'm not talking about that yet let
19:40
me let me finish this first come on man
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come on I'm like in the middle of doing
19:47
homework all right so what's the X and Y
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intercept right x and y intercept so the
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x intercept is found when the function
19:57
is Crossing the what the xais and the Y
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intercept is where the function crosses
20:02
the Y AIS so to find x intercept you
20:06
replace y by zero and to find the uh Y
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intercept you replace X by 0 this is
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just that simple now you can use to
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graph a line you only need what two
20:16
points right so to graph this line using
20:18
the X and the Y intercept what we're
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going to do is this first let's find x
20:22
intercept right x intercept is that two
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so to find the x intercept we replace y
20:28
what 0 right so it's going to be
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2x minus pretty much 3 * 0 + 8 is equal
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to Z so this is basically
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2x = 8 and / 2 get x = -2 now the way
20:46
you write the x incept is this X is4 Y
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is 0 you have to write this as a ordered
20:52
pair you have to be careful how you
20:54
write that that's an order pair is -4
20:56
and0 right this is how you write the x
20:59
intercept now to find the Y
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intercept you do the same thing we going
21:07
to replace X by 0 and solve for
21:11
y unfortunately here we going to have a
21:14
fraction right this is
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gone we get -3 y =
21:20
8 and /
21:23
by3 Y is 8/3 right so the Y intercept
21:28
therefore is
21:29
zero and /3 now you can use this two to
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graph the function so if you go
21:36
here you're going to put -4 1 2 3 4
21:40
that's -4 and zero that's the x
21:42
intercept and the Y intercept is 0 and
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83 83 is about 2. what there put the
21:50
calculator
21:54
8
21:55
2.6 all right
21:58
1 2 3 4 5 6 so two points is somewhere
22:02
around here right so this function
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therefore is going to look like this as
22:07
you can see that's a straight line right
22:09
it's a straight line and it's a linear
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what's the what's the domain and the
22:13
range of this
22:15
function what's the domain to negative
22:19
infinity and infinity what's the range
22:21
negative Infinity it is it one1 yes
22:25
right this is a one1 function domain is
22:28
Infinity to positive Infinity all right
22:31
that closes the section so now you can
22:33
start on your homework so many help
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