Hi, guys.
Welcome to Algebra 1.
Today's lesson's going to focus
on solving literal equations.
If you think back
to pre-algebra,
you had practice
solving equations.
You'll use that knowledge
to get you through this one.
You ready? Let's go.

(Describer) She uses a stylus on a screen.

Okay, before we get
into literal equations,
what they are,
think back to linear equations.
In pre-algebra, you probably
solved something like this.
You learned steps
to get through this.
You'd see, "solve for x."
You've got 5x equals 20,
so you think,
when solving equations,
the goal is to isolate
the variable.
You want to get
the variable all alone.
You have 5 times your variable,
5 times x.
You have to do the opposite
or the inverse operation--
in this case,
of multiplying.
The opposite of multiplying
by 5 would be to divide by 5,
so you'd divide both sides
of your equation by 5.

(Describer) She writes.

On the left, the 5s
would cancel out,
so you'd be left with x equals--
and then 20 divided by 5 is 4.
In pre-algebra you learn
to solve one-step equations.
We can solve them
in one step.
Okay? All right.
Continuing back on memory lane,
let's look at this.

(female describer)
Title: Solve for x.
2/3x equals 4.

(Describer) Title: Solve for x.
Two-thirds x equals four.

(teacher)
You could solve this
in one step or two
depending on how you thought.
I'll show you how
to solve it in two.
It probably alarmed you
that it had a fraction,
but we're not afraid.
I'll show you how
to get rid of it.
Here I have
2/3x equals 4.
I want to get rid
of that fraction.
That would probably
make us happy.
Let's move this
a little bit.
Look at the fraction
separately.
Consider its numerator;
consider its denominator.
We want to get rid
of the denominator first.
Two-thirds--if you thought about
that like a division problem,
it's 2 divided by 3, and
we're multiplying that times x.
I'm dividing by 3; thinking
about this as 2 divided by 3.
What's opposite
of dividing by 3?
Multiplying by 3, right?
That's how you'll start
to get rid of this fraction.
I'll multiply the left side
of my equation by 3.
Put my multiplication symbol
and a 3.
Whatever you do to one side
you do to the other.
I multiply
the right side by 3.
On the left side, if you
remember, your 3s cancel out.
All you're left with
is 2x,
so 2x equals--
On my right side,
I have 4 times 3.
I know 4 times 3
is 12--there.
We're at this point where
we have 2x equals 12.
We feel comfortable with that;
it's similar to the last one.
You look here--
the x2 is multiplying x.
What's the opposite
of multiplying by 2?
Divide by 2, right?
You would divide both sides
by 2, your 2s would cancel out,
so on the left
you're left with x equals,
and then 12 divided by 2 is 6.
You'd see x equals 6.
I know in pre-algebra
you handled linear equations.
Now look at this.
Here our equation is A equals
b times h, and solve for h.
When we have
a situation like this,
we're solving
a literal equation.
What's the difference
between a literal equation
and a linear equation like
the ones we were handling?
For literal equations,
your answer is not a value.
It's not going to be
x equals 4 or x equals 2.
Your answer is some other
algebraic expression,
something else basically
with variables still.
Let me work you
through this one.
Here I'm solving for h,
and I have a equals b times h
because when I see terms
together in algebra,
multiplication is holding
them together.
I need to isolate h here;
h is what I'm solving for.
Pretend they're just numbers;
you use the same process.
I have b times h; what's the
opposite of multiplying by b?
Divide by b.
We don't know what b is,
but treat it the same way
we treat a number
like the other equations.
We divide the right side by b.
We're going to divide
the left side by b.
On the right side, "b"s cancel
out and you're left with h.
You have h,
we have equals,
and on the left
we have a divided by b.
You're done, for the most part;
you've solved this.
It makes us feel
more comfortable
if we see
the variable
on the left and the answer
on the right.
You might
reverse this.
I'll say our answer
is a divided by b equals h,
or h equals
a divided by b.
Either one
is acceptable to write.
You might feel comfortable
seeing the variable
on the left and the expression
on the right,
but they're both fine.
You solved your
first literal equation.
You see what I mean--
this a divided by b.
It's not a numerical value.
We don't know values
of a and b; they're variables.
That makes literal equations
different from linear equations.
You may feel like
you're not done,
but as long as you solve
for the variable, you're done.
Let's look at example two.
Here we're solving for r
and we have v
equals pi r squared h.
A couple of things about this
problem might have alarmed you.
First of all,
you probably noticed the pi.
I'm sure you've learned that
you can approximate pi at 3.14.
It's a very long
number that never ends
and never repeats and goes on
and on for infinity.
Your calculator
will usually take it
10 or 11 decimal places, but
generally we round it at 3.14.
The further you get
in your math classes--
Algebra 1 and Geometry--
the further you go,
you'll see "pi" more often
than you'll see 3.14.
When you see it, use it;
don't feel you have to use 3.14.
Go ahead
and keep going with "pi."
Another thing, you probably
noticed we have an exponent.
That adds to our steps
when we're solving this problem,
generally the same process
we're doing every time.
Let's jump in.
Let me move this over so I have
a little more space to work.
I have v equals pi
times r squared times h.
I have the product
of three terms on my right.
I know I'm solving for r,
so my goal is to isolate r.
Right now it's r squared, so
I'll work on getting that alone.
If these are being
multiplied together,
I ask myself: What's
the opposite of multiplying?
Dividing.
I want r squared to stay,
but I want everything
else to leave.
I divide by what
I want to get rid of.
I divide by pi h--
by that product.
Whatever I do to one side
I do to the other,
so I divide
the left side by pi h.
I've got that
taken care of.
On the right side your pis
cancel, your "h"s cancel,
so you're left with r squared.
And on the left side,
you're left with v
divided by pi h.
What is v?
We have no idea.
What is h?
We don't know.
That makes it literal,
because you're working with
a lot of variables
in these kinds of equations.
I have v divided by pi h
equals r squared.
I'm trying to solve for r,
not r squared.
I ask myself: What's the
opposite of squaring a number?
I need to undo the squaring.
And it's square root.
We take the square root
of r squared
and the square root
of v divided by pi h.
That allows us get at r,
what we're trying to solve for.
We'll take the square root of
r squared and the square root
of that quotient
over there.

(Describer) She writes radicals.

The square root of r squared--
I know that's r.
V divided by pi h--
that square root,
I write it just like that
because I don't have
numerical values for v and h.
The square root of v
divided by pi h--
People are often
more comfortable
seeing the variable
they were solving for
on the left and then
the answer on the right.
So you could
rearrange this.
You could write this as
r equals square root--
big square root--
v divided by pi h.
Either are acceptable answers,
just whichever one you feel
more comfortable writing.
Let's try another.
See if you're feeling
more comfortable with these.
We have v equals 1/3bh.
They're asking us
to solve for b.
We've got a fraction,
but we practiced with that,
so we're not alarmed, right?
We're trying to get b,
to solve for b.
We'll feel better if we can
get rid of the fraction.
Consider the denominator first.
My denominator is 3.
If you think about that
fraction, like 1 divided by 3,
that will give you a hint
of your inverse operation.
If you're dividing by 3--
What's the opposite
of dividing by 3?
Multiplying by 3, right?
I multiply the right side by 3
and the left side by 3.
On the right side,
that allows my 3s to cancel
and I am left with 1bh.
Because 1 is not going to change
anything as a coefficient,
I am going to write bh.
Then I have 3 times v,
so I have 3v.
I'm not done
because I'm trying to get b.
I have 3v equals b times h,
so h is multiplying
times the b.
So what's the opposite
of multiplying by h?
Dividing by h, right?
Divide the right side by h;
divide the left side by h.
And what happens on the right
is your "h"s cancel out.
You're left with B.
On the left,
you have 3v divided by h.
And you are done--
got through that.
It took us about two steps.
Okay? All right.
It's your turn, guys.
Press pause
and take a few minutes
and work
through these problems.
Notice the variable that
I'm asking you to solve for.
When you're ready to check
your answers, press play.

(Describer) Number One: Solve for b.
A equals one-half bh.
Number Two: Solve for w.
V equals Lwh.

(female describer)
1. Solve for b:
A equals 1/2 bh
2. Solve for w:
V equals lwh.
Okay, let's see how you did.
Get the pointer tool back.
On the first one,
you're solving for b.
I got b equals
2a divided by h.
On the second one,
we were solving for w.
I got w equals
v divided by lh, okay?
I'll show you how I got those;
I'll show my work.
Let's get the pen;
okay, here we go.

(Describer) Number One:

I was solving for b.
Let's move things out of the way
and get room to work.
Let's get that
out of the way,
and let's move this up.
There we go.
I want to get b by itself,
so I look at my problem
and see
what's going on with b.
I think I'll scoot
that over more.
First, I spot that fraction.
Get that out of the way.
Think about that fraction
as 1 divided by 2.
That hints that,
to get rid of it--
or you'll hear people
say "clear" their fraction--
I need to multiply
both sides by 2.
On the right,
my 2s cancel out,
and I'm going to write bh,
because that 1 as a coefficient
isn't changing anything.
And then on the left
I have 2 times a. That's 2a.
I'm trying
to solve for b.
I have 2a
equals b times h.
The opposite of multiplying
by h--divide by h.
Divide the right side
and the left side by h.
On the right,
your "h"s cancel out.
So you're left with b.
And on the right,
2a divided by h.
You noticed, I reversed it
when I wrote the answer.
This one is
acceptable also, okay?
Let me show you how
I got that next one.
Let's move some things
out of our way.
Move that. This time
they're connected, okay.
I'll rewrite that so I won't
get that too crowded.
V equals l times w times h.
I'm solving for w.
What's going on here?
I have v equals
l times w times h.
I want to get w by itself.
It's being multiplied
by l and h.
What's the opposite of
multiplying by l and h?
Divide.
I'll divide the right side
by lh
and then the left side.
What I do to one side,
I do to the other.
On the right, my "l"s cancel,
my "h"s cancel,
and I just have w.
On the left, v divided by lh.
On the answers,
I reversed these,
but this is fine too.
Whichever way you want
to write it, okay?

(Describer) V divided by Lh equals w.

All right, guys.
I hope you feel confident
about solving literal equations.
I hope to see soon
for more Algebra 1. Bye.

(Describer) Accessibility provided by the US Department of Education.