Hi guys.
Welcome to Algebra I.
Today's lesson focuses on
representing a direct variation
algebraically and graphically.
Your prior knowledge about
the direct variation equation
and graphing
direct variation equations
will help you
get through this lesson.
You ready to get started?
Let's go.
To jog your memory
before we dive into this topic,
recall what the equation of
a direct variation looks like.
There are two equations we can
work with with direct variation.
Either y equals kx
or k equals y over x.
Depending on the problem
we're working with,
sometimes one equation
more helpful than the other.
Remember that k was your
constant variation, right?
Keep that in mind.
Let's keep going.
Let's look at this one:
example 1.

(female describer)
Within a set of braces
are three ordered pairs:
minus 2 and minus 16,
3 and 24, and 4 and 32.

(teacher)
Write an equation
for the direct variation
represented in the relation.
In this situation,
because we want to write
that equation
and we're
not trying to determine
if this is
a direct variation--
we're not testing it out--
we're going to start
by using--
We'll try to get
to this.
Let me scoot that
so you can see.
Our goal is to get
to an equation in this form,
y equals kx.
What we need to determine is,
what is k?
Because we are told already
that this is a direct variation,
we don't need to test out
those quotients
like we do for each
of these ordered pairs.
We don't need to do negative 16
divided by negative 2,
24 divided by 3,
and then 32 divided by 4
to ensure we get
the same answer.
We're told it's a direct
variation situation,
so we know
we'll get the same answer.
If you don't believe me,
I'll show you
just so you know.
Negative 16 divided
by negative 2 is 8;
24 divided by 3, it's 8;
and 32 divided by 4--
let's get
a little space here.
It's 8, okay?
When you're told that you have
a direct variation situation,
don't test
your ordered pairs.
You really just need
to pick one--one ordered pair.
What you want to find is,
what is k?
To write this equation,
this y equals kx,
I need to figure what is k?
What is
the constant variation?
I proved already that this was
a direct variation situation--
my testing out
the ordered pairs--
so you can tell
that k is 8.
If k is 8 for this situation,
then I can write the equation
as y equals 8x.
And I'm done, okay?
To solve these problems
where you write the equation,
all you need to do
is figure what is k
and fill it in
to that general format,
that y equals kx,
all right?
Let's look
at another one here.
We have a table, but
the directions are the same:
"Write an equation
for the direct variation
represented in the relation."
Kind of a mouthful.

(female describer)
The ordered pairs in the table
are minus 20 and minus 5,
28 and 7, and 44 and 11.

(teacher)
What we want to figure is,
what is k?
Then we can write
our direct variation equation.
Because I'm already told this
is a direct variation situation,
I don't have to test
every ordered pair.
I'll just pick the first one.
In order to figure out
what k is, I need to do--
I'll write that here--
y equals k divided by x--
I'm sorry--
k equals y divided by x.
For this one, k, if I pick
the first ordered pair--
but you can choose
any ordered pair--
y is negative 5;
x is negative 20.
I simplify this
by dividing my numerator
and my denominator
by negative 5.
Negative 5 divided
by negative 5, positive 1.
Negative 20 divided by
negative 5 is positive 4.
My k for this situation
is 1 over 4.
I want to write my equation
in the format of y equals kx.
Now that I know that k is 1/4,
y equals 1/4x.
Okay? And you're done.
You have your equation
for this direct variation
situation, okay?
If you chose to write
your equation in this format--
the k equals y divided by x--
that's still acceptable.
But when we're dealing
with a table
or set of ordered pairs,
and eventually
we get to the graphs,
you want it in the y equals
form most of the time.
Both of these are acceptable
if you were to have--
since your k is 1/4,
you could say that 1/4
equals y divided by x.
That is also true
for this situation.
Get in the habit
of figuring what k is
and then filling it in
to that style of the equation.
Okay, let's keep going here.
I've got a couple for you.
You have a set of ordered
pairs and a table;
you're asked to write
that direct variation equation.
Remember, figure what k is
and then fill it in
to that y
equals kx equation.
To check your answers,
press play.

(describer)
Relation A is a set
of the three
ordered pairs in braces--
2 and 6, 5 and 15,
and 9 and 27.
Relation B
is three ordered pairs
in a table with two columns,
headed x and y.
The ordered pairs are 8 and 4,
10 and 5, and 18 and 9.

(teacher)
Let's see how you did.
For A--
Let's move this
out of the way here.
You should have got
y equals 3x for your equation.
For B, your equation
was y equals 1/2x.
Want to see
how I got these?
Stay with me,
and I'll show you.
Look at the first one.
I first need
to figure what k is.
k, I can figure out--
let's get the pen--
by determining a quotient,
y divided by x.
Pick any ordered pair.
I'll pick the middle one.
I know that's my x;
that's my y--
k would equal
15 divided by 5
for my y divided by x,
which is 3.
I want to write my final
answer in the y equals kx form.
Now that I know
that k is 3,
y equals 3x.
You got it;
you're done on that one.
Figure out k and drop it in
to that y equals kx equation.
Let me show how I got that
second one--my table here.
Again, just figure
out what k is first--
k equals y divided x.
I picked
the last ordered pair.
You know this
is a direct variation,
so it doesn't matter
which pair of points you pick.
I see 9 is the y
and 18 is the x.
I can simplify this fraction
by dividing
both the numerator
and denominator by 9.
9 divided by 9,
that's 1;
18 divided by 9,
that's 2.
So, k is 1/2, right?
Scroll down a little bit.
I know I'm trying to fill in
to the y equals kx equation.
Now that I know k is 1/2--
a little more room--
y equals 1/2x.
And you got it, okay?
That's how I got those two
answers for those two questions.
Let's look at a word problem.
You knew it was going
to pop up eventually.
"Example 3: The amount of money
Jasmine earns varies directly
"with the number of hours
she works at the library.
"Last week, Jasmine worked
12 hours and earned $111.
How much will she earn next
week if she works 15 hours?"
Okay, now, there's
some steps we can follow
with a word problem
like this,
where we're given a set of
information and then asked
to actually generate another
value for this situation, right?
Here are the steps
to take:
The first thing is,
we're going to determine
what's x, what's y, what's k.
We get those important values.
Then we write
the direct variation equation.
We have all the information
needed to write it.
After we've got
that equation, we use it
to answer the question--
to figure out that value
they want you to get
for that answer.
Remember those three steps:
determine x, y, and k,
write the equation,
then use it
to answer the question, okay?
Back to that example, let's
highlight the key information.
Break this problem
down a little bit.
The amount of money Jasmine
earns varies directly
with the number of hours
she works.
The money varies directly
with the number of hours.
Last week,
Jasmine worked 12 hours--
so 12 hours--
and earned--oop--
there we go--
and earned $111.
How much will she earn
if she works--next week--
if she works 15 hours?
We want to know how much
is she going to earn
working 15 hours, all right?
Step 1, let's figure out
what's x, what's y, what's k.
Let's get
the pen back here.
They're telling us--
I lost
some highlighting--
there we go.
They're telling us, the amount
of money varies directly
with the number of hours.
That means, the amount of money
depends on the number of hours.
That means y, since it's
our dependent variable,
is the amount of money,
and x, because it's
our independent variable,
that's the number
of hours, okay?
We've got the x;
we've got the y--
y is the money,
x is the hours.
We've got to figure k.
That's when we use that initial
information we were given,
that Jasmine worked
12 hours to earn $111,
to be able to figure out what
is k for this situation.
Let's get that.
Get a little more space.
Going to have to scroll
up and down here.
For direct variation,
k equals y divided by x.
We know that y is the money.
They told us she earned $111.
x is the hours.
She earned that
when she worked 12 hours.
Let's get
some more space, okay?
So k will equal that $111
divided by 12 hours, okay?
For this, because this division
isn't so pretty,
we switch to the calculator
to get our answer for this one.
Let's get that calculator out.
I like to clear the memory
because, like we saw there--
clear all the information
so you're starting
with a clean slate.
If we look at our problem
to see what we were trying
to determine,
we want to figure out
what is 111 divided by 12.
We do that division
instead of leaving fractions
because we're dealing
with money.
If we have a decimal, that's
like the coins for the money.
We've used the fractions
and just sticking with that,
but we want the decimal
for this situation.
If you used fractions,
you're not wrong,
but it will just be easier to
see exactly what our k is here.
So 111 divided by 12;
that's 9.25, all right.
Let's get that down,
and we'll interpret
what that means exactly.
Go full screen, okay.
So we know that k is 9.25.
That's our constant.
If we go to step 2 now
and write that equation--
I'm going to number this
"step 2."
Let me scroll up,
and I'll label this "step 1"
just to help us
stay organized here.
Now that we know
k is 9.25,
we fill that k in to our
y equals kx equation,
so y equals 9.25x.
Basically, what that's
telling us is that--
I believe it was Jasmine
we were working with in--
that Jasmine earns
$9.25 per hour.
That's what
that means.
We used that
given information
that after working 12 hours,
she earned $111
to figure that constant rate
that Jasmine is being paid at.
We've got step 2.
We've got our equation,
y equals 9.25x.
Scroll up to see what
we're being asked to figure.
"How much will she earn next
week if she works 15 hours."
That 15 hours,
that's our x.
We've already identified
that x is the number of hours.
Let's scroll back down
so we can do step 3, okay.
We're going
to determine
what is y when x is 15?
We're going to substitute
that value in there for x.
Once we figure out this product,
this 9.25 times 15,
we'll know how much Jasmine
makes after working 15 hours.
Again, this multiplication
here isn't so pretty.
We've got a decimal.
I switch to the calculator
to get us our answer quickly--
9.25 times 15.
"9.25 times 15."
We've got 138.75.
Let's get that down.
Okay.
That was 138.75
is what we got for y.
So that means
that after working 15 hours,
Jasmine will
earn $138.75.
That's the answer
to that question.
To bring it together,
I'm going to scroll again.
Scroll up
to the beginning of our work.
We started out
reading the problem.
We highlighted those
key parts of information
in order to determine what
was x, what was y, what was k.
That was our step 1.
After we figured out
that k was 9.25,
we wrote our equation
that y equals 9.25x.
We used that equation
to answer the question,
to figure out
after working 15 hours,
how much money Jasmine earned.
We determined
that it was $138.75, okay?
Keep those steps in mind,
and let's try
another word problem here.
Okay, let's do a little
moving around here.
"Example 4: Julio subscribes
to an online streaming service.
"The monthly cost
varies directly
"with the number
of movies streamed.
"Two months ago, Julio paid $18
to watch 4 movies.
"If Julio's total cost
last month was $27,
how many movies did he watch?"
We use our same strategies
as we did before.
We've read
through it once.
Now let's highlight
that key information
so that we can determine
what's x, what's y, what's k.
Okay, we know Julio subscribes
to an online streaming service.
"The monthly cost
varies directly
with the number
of movies streamed."
The monthly cost varies directly
with the number of movies.
So two months ago,
Julio paid $18
to watch 4 movies.
If Julio's
total cost last month--
so cost last month--was $27,
how many movies did he watch?
We've got that
key information.
Let's use our steps
to get to the answer.
Step 1, we've got to figure out
what's x, what's y, what's k.
Let me switch
to my pen.
We were told in the problem
that the monthly cost
varies directly
with the number of movies.
The cost depends
on the number of movies.
So y...
that is our
dependent variable,
and that's the cost.
And x, that's our
independent variable.
That's what we can control.
We can control
the number of movies.
We'll say,
number of movies, okay.
We've identified
the y and x.
We've got to get k.
Let's look
back up here.
We'll use that first set
of information
that Julio paid $18
to watch 4 movies, right?
Let's scroll down.
We're doing
a lot of scrolling
because we've got
a lot of work.
To figure out k,
we know that--oop--
that's step 1--
we know that k is y
divided by x, right?
y is our dependent
variable, the cost.
If you look up here,
it said, Julio paid $18
to watch 4 movies.
So the cost, 18...
number of movies, 4, right.
Get more space.
I go to the calculator
so I can get that rate,
18 divided by 4.
Let's switch over here;
let's clear that out.
18 divided by 4, enter--
4.5, okay.
We've got our k.
Let's go back
and write it down.
All right, scroll down.
Get some more space.
So k equals 4.5.
Because we're dealing
with money here,
that 4.5--
that's basically $4.50.
I put the 0
to stay consistent
because we're dealing
with money in this case.
If you had that k was 4.5,
you're not wrong for that.
You're right, okay.
Now that we know
that k is 4.5,
we can fill it in
to that direct variation
equation--
Get some more room.
I'm going to fill in
that k equals 4.50--
I use the 0--x.
To interpret,
all that means is that Julio
is paying $4.50 for every movie
that he is streaming.
That's all that this means,
that equation, okay?
We've got step 2;
scroll up to see
what we're being
asked to find.
The total cost
last month was $27,
so how many movies
did he watch, okay?
They've given you
the cost.
They have given you y,
and they want you
to determine what is x.
How many movies
did he watch? Okay.
Let's look
at exactly where
we're going to replace
those values.
Last month, the cost was $27,
and y represents the cost.
So step 3,
I'm going to have 27
equals 4.50--
4.50 times x.
So notice this time,
I replaced
that given value for y.
They gave me
the cost,
and we already determined
that y was our cost.
This time, we know
the cost was $27.
We want to determine
how many movies did Julio watch
if that's
what he paid.
We have this
one-step equation,
27 equals 4.50 times x.
We will divide
both sides by 4.50.
That division isn't so pretty,
so get the calculator out--
27 divided by 4.50--
27 divided by 4.50...
6, okay.
So we got--for this situation,
we go back to our work.
Let's scroll down--I'm going
to need some more space.
That 27 divided
by 4.50 equals 6.
What this means
is that if Julio paid $27,
then he watched
6 movies, okay?
Let's scroll back up
just to bring that together.
Make sure you got the steps.
After you've read through
the problem and highlighted
that key information,
start by identifying
your x and your y and your k.
Figure your independent
and dependent variables
so you can figure out
what k is.
After you have k, go ahead
and write the equation.
Use it to figure exactly what
you are being asked
to solve for to get that value.
Those three steps will get you
through these word problems.
I want you to use those
to solve this one, okay?
Let me get a few things
out of your way so you can see.
Press pause
and take a few minutes
and work through this problem.
To check your answer,
press play.

(describer)
The cost of Suzanne's
data plan varies directly
with the number
of gigabytes she uses.
Last month, Suzanne used
6 gigabytes of data
and was charged $22.50.
If Suzanne uses 8 gigabytes
of data this month,
how much will she be charged?

(teacher)
All right,
let's see how you did.
Let's read this.
"The cost of Suzanne's data plan
varies directly
"with the number gigabytes
she uses.
"Last month, Suzanne used
6 gigabytes of data
and was charged $22.50."
My highlighter out--
"If Suzanne uses 8 gigabytes
of data this month,
how much will she
be charged?"
Let's start
highlighting.
So the cost
varies directly
with the number
of gigabytes she uses.
Last month,
she used 6 gigabytes
"and was charged $22.50.
"If Suzanne uses
8 gigabytes of data this month,
how much will she be charged?"
We've pulled out
that key information here.
Let's start with our first step
to determine what's x,
what's y, what's k, okay?
Get some work space here;
I have to do some maneuvering.
First, it tells us
that the cost varies directly
with the number
of gigabytes she uses.
The cost depends on the number
of gigabytes she uses.
That means the cost
is the dependent variable.
Abbreviate it.
The number of gigabytes
is the independent variable.
That's what she can control.
Gigabytes, okay.
That means that--
I need to scroll again.
Last month,
she used 6 gigabytes
and was charged $22.50.
We're going to use that
information to get k, okay?
Move that up.
See if we can see
so we don't have to scroll.
Still on step 1.
So k equals y divided by x.
They told us,
she used 6 gigabytes
and was charged $22.50.
So y is the cost.
The cost, $22.50,
and x is 6 gigabytes,
the number
of gigabytes that she uses.
We need to determine,
what is the value
of 22.50 divided by 6.
Let's go to our calculator--
22.50 divided by 6.
22.50 divided by 6...
3.75.
Let's write that down.
Get some more space;
here we go.
So k equals 3.75.
That is our k.
We've got k.
We've got y. We've got x.
We go to step 2
and write the equation.
Because we know
it's y equals kx
and we know that k is 3.75,
y equals 3.75x.
Basically, Suzanne is charged
$3.75 per gigabyte she uses.
That's what this
equation is telling us, okay?
Now we're ready for step 3.
Let's see what they
are asking us.
"If Suzanne uses 8 gigabytes,
how much will she be charged?"
Replace 8
in the right spot--
substitute 8
in the correct location--
8 gigabytes--gigabytes
is my independent variable.
That's my x.
Let's scroll down.
I'm going to need
y equals 3.75 times 8.
This time,
they're giving me the x.
They're giving me that
independent variable's value
and want me to figure out y--
figure out that cost.
Let's go to the calculator:
we need 3.75 times 8.
3.75 times 8...
30.
Who knew it would be
a whole number?
Let's go back
to our work here.
We have figured out
that y equals 30.
That means that if Suzanne
uses 8 gigabytes,
she'll be charged
$30 for that.
Good job on that one.
Let's keep moving here.
It's time to attack the graphs.
I move this up
because our directions
are underneath our problem,
underneath our point.
For example 5: "the given point
is located on the graph
of a direct variation."
We have that point
in our second quadrant.
They're telling us, it's part
of a direct variation.
"Indicate another point
contained on the graph
of this direct variation."
I want to jog your memory,
and I want you
to look at this.
Remember that a direct
variation, when you graph it,
it's always a line that
passes through the origin.
It's not always
this exact line.
This is just an example
of one direct variation,
but every direct variation
is a line
that passes
through the origin.
It may have a positive
or negative slope,
but it must be a line that
passes through the origin.
Let's go back to that problem.
They've given us one point
that's on that line
of this direct variation.
They want us to find
another point.
Think about it--
the only point
that you could always
be certain
is on a direct variation's
graph is (0,0)--
right here at the origin.
Because
if you know nothing else,
you know that
for a direct variation,
that line will pass
through the origin graphed.
You just need
to plot your point there
at the origin,
and you're done.
That's the only certain point
on the graph
of this direct variation.
Okay?
Let's keep going here.
Look at this--
it's your turn.
"Show an example of the graph
of a direct variation equation.
Indicate
at least two points."
There are several
correct answers
to this question here,
okay?
Figure out
where are two points
that would be on
this direct variation.
One certainly
has to be there,
and there another one
that's up to you.
So press pause.
Get graph paper out.
Get this one done.
To check your answer,
press play.
Let's see how you did.
Now, if this
is a direct variation,
I know, if nothing else,
there is a point here
at the origin.
That's the one
you knew would be there.
The other point you plotted
could be any
of these four quadrants.
Maybe you put a point here.
That would have been fine.
Maybe you didn't
put a point there.
Maybe--erase that.
Maybe you put a point
over here.
That would have been correct.
Or maybe
you didn't put a point there.
Maybe you put a point
way over here.
Where your other point was
was completely up to you.
If it was in these
four quadrants, you were good.
But you had to
have one point at the origin
in a direct
variation situation.
All right.
Great job, guys,
working on these problems,
representing direct
variations algebraically
and graphically.
Hope to see you soon
for more Algebra 1. Bye.