Hey, guys.
Welcome to Algebra I.
Today's lessons
are going to focus on
representing inverse variations
algebraically.
Your prior knowledge
about what the equation of
an inverse variation looks like
will help you
get through this lesson.
Ready? Let's go.

(Describer) She gets a stylus.

Just to jog your memory,
let's look at this example
where you're asked
to determine:
Is this relation
an inverse variation?

(female describer)
On a screen, a table
has two columns,

(Describer) On a screen, a table has two columns, one labeled x, one labeled y. Vertically in the x-column are minus 1, 5 and 10. Vertically in the y-column are minus 5, 1 and one-half.

one labeled x,
one labeled y.
Vertically, in the x column,
are -1, 5, and 10.
Vertically, in the y column,
are -5, 1, and 1/2.
If you remember,
for an inverse variation,
k was a product
of y and x, right?
That was that constant
you were looking for every time.
If you went through
your table and saw
that each product for each
x and y pair was the same,
you knew you had
an inverse variation situation.
So, we tested this one out:
-1 times -5, that's 5.

(Describer) She writes.

5 times 1, that's 5.
10 times 1/2--
Let's step off to the side.
10 times 1/2--
I'll make that a 10 over 1
to see it as a fraction.
10 times 1 is 10,
1 times 2 is 2,
10 divided by 2 is 5.
I see that I do have
the same product
for each ordered pair
in this table.
So, I could say, yes,
that this is
an inverse variation situation.
Okay? Keep that in mind.
Jog your memory
a little bit more here.
Okay. If you remember--
Let me get
that out of your way.

(Describer) She changes the screen.

You could represent
an inverse variation
by two different equations:
y equals k over x,
or k equals x times y,
or y times x,
because you could multiply
in any order that you want.
Feel free to swap
those variables, the x and y.
These are the formats
of your inverse
variation equations,
and you know that k
is your constant of variation.
How we're gonna step
the problems up a notch
is we're actually gonna write
that inverse variation equation.
Keep this in mind,
and let's look back
at this problem.
This is the same table
we had a second ago,
when we determined
that the constant of variation--
let's get the pen back--
was indeed 5.
We found that
each of these products was 5.
We knew this
was an inverse variation.
What that means
is that k equals 5.

(Describer) She writes.

And if we want to write
an equation
to represent
this inverse variation,
then you just need to know
that y equals k over x,
or k divided by x.
And if I know now that k is 5,
then I can say
that y equals 5 over x.
And I have written
an inverse variation equation
for this situation.
Okay?
Keep that in mind,
and let's do that one more time,
but this time, we'll look at
a set of ordered pairs.

(Describer) The ordered pairs are inside braces. Each pair is in parentheses and has a comma between them. They're 2 and 36, 6 and 12, and 8 and 9.

(describer)
The ordered pairs
are inside braces.
Each pair is in parenthesis
and has a comma between them.
They're 2 and 36,
6 and 12, and 8 and 9.

(instructor)
Again, we're asked
to write an equation
to represent
this inverse variation,
Because
they've already told us
that this is
an inverse variation,
I don't have to test out
each of these products.
It's guaranteed to me
that this is
an inverse variation situation.
All I need to do is pick
one of these ordered pairs
and use them to figure out
what my k is,
because I know k
is constant for this.
I'm gonna pick the smallest
numbers of ordered pairs
so I can use mental math
to figure it out.

(Describer) Eight and nine.

I know that k is x times y,

(Describer) She writes.

or y times x,
however you like to write it.
So, 8 is x, 9 is y.
So, k equals 8 times 9,
which is 72.
If I'm writing
an inverse variation equation,
I can write,
if I fill into
the y equals k over x style--
scroll down a little bit--
then I could write
y equals 72 over x,
and I'd be all done.
If you didn't want to write
your equation in this style,
and you wanted to write it as
k equals x times y--
I'm gonna rewrite this
so you get a good visual
of what I mean.
Because you know k is 72,
72 equals x times y.
That's a valid equation also.
That is also
how you could represent
this inverse variation.
Generally you'll see it
in this style
where it's solved for y,
but if you wrote it in this way,
you're not wrong.
Just be familiar
with both of these,
but commonly
you'll see the equation
where it's already
solved for y.
Let's keep moving.
And it is time for you
to try a couple.
You're given a set
of ordered pairs and a table,
and you're being asked
to write the--
actually not
the direct variation equation,
you are being asked
to write the inverse.
Let's get that changed.
The inverse variation
represented in
the set of ordered pairs
and in the table.
You got to figure out
what your k is,
and then fill in
to that y equals k over x style.
When you're ready to compare
your answers with mine,
press play.

(Describer) Relation A is a set of ordered pairs: 1 and 12, 3 and 4 and 6 and 2.
Relation B is a table. Under the x and y columns respectively are 2 and 9, 3 and 6 and 6 and 3.

(describer)
Relation A is a set
of ordered pairs,
1 and 12, 3 and 4,
and 6 and 2.
Relation B is a table.
Under the x and y columns
respectively are
2 and 9, 3 and 6,
and 6 and 3.
Let's see how you did here.
For A--
let's get this
moved out of the way--
you should have gotten
y equals 12 over x,
or y equals 12 divided by x.
And for B,
y equals 18 divided by x.
If you want to see
how I got that,
I'll show you.

(Describer) She changes the screen to just A.

For the first one,
remember, the first thing
you have to do
is to figure out what is k.
Because you were told
that these were
inverse variation situations,

(Describer) She writes.

you know that k
is gonna equal x times y.
Pick any pair you want
for x and y,
and then determine
what your k is.
I'm picking the first pair.
It's the easiest one
since I'm multiplying by 1.
K would equal 1 times 12,
which is 12.
Right?
If I'm filling into
the y equals k over x format,
and now I know that k is 12,
then I can say
y equals 12 over x.
And I'm done.
I've gotten my equation
to represent
this inverse variation.
Okay? Let me show you
how I got B.
Had a table here.
So again, same process.
First get k.
K is a product of x and y.
Pick any pair
because you've already been told
this is
an inverse variation situation.
I'll pick the last pair
this time, 6 and 3.
So, k would equal 6 times 3,
which is 18, right?
And if I'm gonna fill into
the y equals k over x format,
then I would have
y equals 18 over x.
And I'm done.
Okay? It was
as simple as that.
Figure out your k,
and then drop it into
that y equals k over x format.
You know I couldn't
let you out of here
without looking at
a few word problems.
Let's look at this.
I'll get my highlighter.

(Describer) She reads.

"The amount of time
it takes to paint a room
"varies inversely with
the number of painters working.
"If 5 painters
"finished painting a room
in 64 minutes,
"how long would it
have taken 8 painters
to complete the same room?"
When you're presented with
a lengthy word problem,
there are just three steps
that you're gonna want to follow
in order to solve it.
Here are the steps.
First, determine your x,
your y, and your k.
Get those important values.
Then write the equation.
Write that inverse
variation equation,
and then go ahead
and answer the question.
If you follow this process,
you'll be able to work through
these word problems easily.
Determine x, y, and k,
write the equation,
use it to answer the question.
All right?
Let me show you what I mean.
Jump back to example three.

(Describer) The word problem.

I've got my highlighter ready.
Let me start by figuring out
the key information
that'll help me determine
what's x, what's y, what's k.
"The amount of time
it takes to paint the room
varies inversely."
The amount of time
varies inversely
with the number of painters.

(Describer) She highlights.

Okay.
"If 5 painters finished painting
a room in 64 minutes."
So, 5 painters, 64 minutes.
"How long would it have taken
8 painters--"
So, how long would it have taken
8 painters
to complete the same room?
All right. Let's begin
working our process here.
Scroll a little bit.
I'm gonna label this
the first step,
which is to determine
what's x, what's y, what's k.
If we look back at what
we highlighted here at the top,
it tells us
that the amount of time
varies inversely
with the number of painters.
So, the time depends on
the number of painters.
All right?
So, y is
the dependent variable,

(Describer) She writes.

in this case,
and it's the time,
and x is
the independent variable,
because we can control
the number of painters.
Lock in that order here.
Let's erase that.
And let's write that y
is the independent variable.
And that's
the number of painters.
That's what we can control here,
how many people
are actually painting the room.
We know y is the time,
x is the number of painters,
so we can figure out
what k is for this situation.
They told us that it took
5 painters 64 minutes.
I'm gonna use that information
to figure out k.
Get some space here.
So, k equals x times y,
and it took 5 painters
64 minutes.
So, 64 minutes is my y,
and 5 is my x.
I noticed
a little something up here too.

(Describer) She erases a misspelling of "independent".

Let's get that fixed
to "independent."
There we go. Make sure
that's spelled correctly.
All right.
So, 5 painters, 64 minutes.
So, 5 painters, that's my x.
64, that's my y,
and I just need to do
this multiplication.
Because these numbers
are large,
I'm gonna switch
to the calculator
to make
the multiplication easier.
So, we need 5 times 64.
Let's clear my memory here.
Just like to do that
whenever I first start working
on the calculator.
5 times 64.
We have 320. Okay?
So, back to our problem.
Let's scroll down here.
K equals 320.
We've gotten step one
taken care of.
We know the values of x and y,
we know what they represent,
and we know the value of k.
We can go ahead and write
our inverse variation equation
now that we know what k is.
Step two--because we're gonna
fill into y equals k over x,
and we know that k is 320--
then we'll have
y equals 320 over x.
Okay? And we've got
step two handled.
We've written
our inverse variation equation.
Now it's time to scroll up
so we can use that equation
to actually answer the question.
How long will it have taken
8 painters
to paint the same room?
We know that
the number of painters--
that's our x,
and this time x is 8.
All right?
For step three,
y equals 320 divided by 8.
All right?
I think I can handle
this division here.
320 divided by 8, that's 40.
What that means
is that for 8 painters
in that same room,
it would have
only taken them 40 minutes
to paint that same room.
All right?
And you have solved your first
inverse variation word problem.
I'm gonna scroll back
through it
to bring it
all together for you.
After reading through it,
we highlighted
that key information.
Then our first step
was to identify
what x and y represented
and then used the information
they gave us to solve for k.
We figured out
that k was 320.
Then we wrote
our inverse variation equation.
We used
the y equals k over x format.
Y equals 320 divided by x,
and then they asked us
to determine
how long would it have taken
8 painters to paint that room.
8 was our value for x,
and 320 divided by 8 is 40.
It would have taken 8 painters
40 minutes to paint
that same room.
Keep that process in mind,
and let's try
another one together.
Let me do
a little moving around here.
Let's move that over there
and get the highlighter ready.
We know what's coming, right?

(Describer) She reads.

"The length of a guitar string
varies inversely
"with the frequency
of its vibrations.
"If an 8-inch guitar string
"vibrates at a frequency
of 628 cycles per second,
"find the length
of a guitar string
that vibrates at a frequency
of 314 cycles per second."
Okay. We read through it.
Now let's highlight
that key information.
"The length
of a guitar string--"
"The length of a guitar string
varies inversely
with the frequency
of its vibrations."

(Describer) She highlights.

"If an 8-inch guitar string
"vibrates at a frequency
of 628 cycles per second,
"find the length
of a guitar string
that vibrates at a frequency
of 314 cycles per second."
So, we've got
the key information highlighted,
so now let's start
working our process,
and step one
is to determine x, y, and k.
All right.
If we look back
at our problem,
we see that the length
of the guitar string
varies inversely with
the frequency of the vibrations.
Because we know,
in an inverse variation
situation,
we're always told
that y varies inversely as x.
So, I can tell here
the length of the string,
that's y.

(Describer) She writes, "y: length of string".

And the frequency, that's x.

(Describer) She writes, "x: frequency."

Now that we know that y
is the length of the string,
x is the frequency,
we can determine
the value for k.
We know that for
an inverse variation situation
that k equals x times y.
Right?
So, we're told here that
an 8-inch guitar string
vibrates at
628 cycles per second.
So, 8-inch, 628, okay?
Those are our values
for x and for y.
So, k would equal--
the frequency
for that situation
was 628 for the 8-inch string.
It's gonna be
pretty large numbers.
I'm going to the calculator
to do this multiplication.
628 times 8.
All right.
So, let's go here.
628 times 8.
And we have 5,024.
I knew it was gonna be big.
All right.
So, let's go back here.
All right.
So, k equals 5,024.
All right.
So, now step two.
Because we know
that for our inverse
variation situation
y equals k over x,
I can represent this situation
by y equals 5,024
divided by x.
Right?
Now we're ready for step three
because we've gotten
our equation down.
Let's see what
we've been asked to find here.
"Find the length
of a guitar string
that vibrates at a frequency
of 314 cycles per second."
Our frequency here,
or our x, is 314.
Let's scroll down.
Okay.
And then step three...
would be y equals 5,024
divided by 314.
Make sure I got
that number right.
That was actually, yep, 314.
Once I do this division,
I'll be able to determine
exactly how long
this guitar string is, right?
That's gonna be
some interesting division,
so I'm going
to the calculator.
Those numbers
are kind of large.
So, 5,024 divided by 314.
5,024 divided by 314.
And we have 16.
All right.
Let's get back
to our problem.
Let's scroll down
so we can write the answer.
16 is what we get for y.
What that means
is a guitar string
that's vibrating
at 314 cycles per second
is 16 inches long.
All right?
And you're all done.
Good job on that one.
Now it's time for you
to try one on your own.
Press pause
and take a few minutes,
work your process,
and work your way
through this problem.
Determine x, y, and k,
write your equation,
and answer the question.
When you're ready to compare
your answer against mine,
press play.

(Describer) Titles: The amount of time it takes to tile a floor varies inversely with the number of people working. If six contractors completed a room in 80 minutes, how long would it have taken 12 contractors to complete the same room?

(describer)
The amount of time
it takes to tile a floor
varies inversely with
the number of people working.
If 6 contractors
completed a room in 80 minutes,
how long would it have taken
12 contractors
to complete the same room?
Let's see how you did here.
"The amount of time
it takes to tile a floor
"varies inversely with
the number of people working.
"If 6 contractors
completed a room in 80 minutes,
"how long would it have taken
12 contractors
to complete the same room?"
Let's highlight
the key information here.
The time it takes,
the amount of time,
"varies inversely with
the number of people working."
"If 6 contractors
completed a room in 80 minutes,
"how long would it have taken
12 contractors
to complete the same room?"
So, we've got
the key information highlighted.
Our first step is to determine
what's x, what's y, what's k.
All right.
Here we have the time
"varies inversely with
the number of people working."
Because we know,
with an inverse variation,
we're given the information
as y varies inversely as x.
That means
that our amount of time is y.
And x is
the number of people working.
That's what we can control.
Okay.
Let's keep going here.
And to get k,
what we're gonna do
is use this first part
of information we were given:
6 contractors, 80 minutes.
Okay.
And because we know
for inverse variation
that k equals x times y,
and I know that,
for this situation,
k will equal--
Gonna need some more room.
I had 6 contractors,
and they took 80 minutes.
So, 6 times 80--
I think I can handle
this multiplication here--
should be 480.
Okay. Your value for k
in this situation is 480.
Let me scroll down
a bit more, get some space.
Now I'm ready for step two,
and I can write
that inverse variation equation.
I'm gonna write it
in the y equals k over x format.
Now that I know
that k is 480--
Let's get more space. Okay.
So, y equals 480 divided by x.
So, I've got that equation.
I've finished step two.
Let's scroll up
so we can answer the question.
"How long would it have taken
12 contractors
to complete the same room?"
So, 12 contractors
has to be a value for x.
That's my number of people.
Let's scroll down here.
So, y equals 480 divided by 12.
Right?
And I think I can handle
that...division.
That should be 40. Okay?
Let me double-check myself
just to be sure here.
Because I've been doing
a lot of mental math.
480 divided by 12
is indeed 40.
All right.
Had that one right.
What that means is,
if we bring this back
to the problem
so we can get the meaning
of this answer
for step three.
Okay.
We were asked to determine
how long it would have taken
12 contractors
to complete that same room.
So, if 12 contractors
had been working--scroll down--
it would have only taken them
40 minutes
to complete that same room.
Okay? Good job on that one.
You have reached
the end of your lesson
on representing inverse
variations algebraically.
Hope to see you back here soon
for more Algebra I. Bye.

(Describer) Accessibility provided by the U.S. Department of Education.