Simple Equations

Let's say we have the equation
7 times x is equal to 14.

Now before even trying to solve
this equation, what I want to

do is think a little bit about
what this actually means.

7x equals 14, this is the exact
same thing as saying 7 times x

-- let me write it this way --
7 times x -- we'll do the x in

orange again -- 7 times
x is equal to 14.

Now you might be able to
do this in your head.

You could literally go
through the 7 times table.

You say well 7 times 1 is equal
to 7, so that won't work.

7 times 2 is equal to
14, so 2 works here.

So you would immediately
be able to solve it.

You would immediately, just
by trying different numbers

out, say hey, that's
going to be a 2.

But what we're going to do in
this video is to think about

how to solve this
systematically.

Because what we're going to
find is as these equations get

more and more complicated,
you're not going to be able to

just think about it and
do it in your head.

So it's really important that
one, you understand how to

manipulate these equations,
but even more important to

understand what they
actually represent.

This literally just says 7
times x is equal to 14.

In algebra we don't
write the times there.

When you write two numbers next
to each other or a number next

to a variable like this, it
just means that you

are multiplying.

It's just a shorthand,
a shorthand notation.

And in general we don't use the
multiplication sign because

it's confusing, because x is
the most common variable

used in algebra.

And if I were to write 7 times
x is equal to 14, if I write my

times sign or my x a little
bit strange, it might look

like xx or times times.

So in general when you're
dealing with equations,

especially when one of the
variables is an x, you

wouldn't use the traditional
multiplication sign.

You might use something like
this -- you might use dot to

represent multiplication.

So you might have 7
times is equal to 14.

But this is still
a little unusual.

If you have something
multiplying by a variable

you'll just write 7x.

That literally means 7 times x.

Now, to understand how you can
manipulate this equation to

solve it, let's visualize this.

So 7 times x, what is that?

That's the same thing -- so I'm
just going to re-write this

equation, but I'm going to
re-write it in visual form.

So 7 times x.

So that literally means x
added to itself 7 times.

That's the definition
of multiplication.

So it's literally x plus x plus
x plus x plus x -- let's see,

that's 5 x's -- plus x plus x.

So that right there
is literally 7 x's.

This is 7x right there.

Let me re-write it down.

This right here is 7x.

Now this equation tells us
that 7x is equal to 14.

So just saying that
this is equal to 14.

Let me draw 14 objects here.

So let's say I have 1,
2, 3, 4, 5, 6, 7, 8,

9, 10, 11, 12, 13, 14.

So literally we're saying
7x is equal to 14 things.

These are equivalent
statements.

Now the reason why I drew
it out this way is so that

you really understand what
we're going to do when we

divide both sides by 7.

So let me erase
this right here.

So the standard step whenever
-- I didn't want to do that,

let me do this, let me
draw that last circle.

So in general, whenever you
simplify an equation down to a

-- a coefficient is just the
number multiplying

the variable.

So some number multiplying the
variable or we could call that

the coefficient times a
variable equal to

something else.

What you want to do is just
divide both sides by 7 in

this case, or divide both
sides by the coefficient.

So if you divide both sides
by 7, what do you get?

7 times something divided
by 7 is just going to be

that original something.

7's cancel out and 14
divided by 7 is 2.

So your solution is going
to be x is equal to 2.

But just to make it very
tangible in your head, what's

going on here is when we're
dividing both sides of the

equation by 7, we're literally
dividing both sides by 7.

This is an equation.

It's saying that this
is equal to that.

Anything I do to the left hand
side I have to do to the right.

If they start off being equal,
I can't just do an operation

to one side and have
it still be equal.

They were the same thing.

So if I divide the left hand
side by 7, so let me divide

it into seven groups.

So there are seven x's here,
so that's one, two, three,

four, five, six, seven.

So it's one, two, three, four,
five, six, seven groups.

Now if I divide that into
seven groups, I'll also want

to divide the right hand
side into seven groups.

One, two, three, four,
five, six, seven.

So if this whole thing is equal
to this whole thing, then each

of these little chunks that we
broke into, these seven chunks,

are going to be equivalent.

So this chunk you could say
is equal to that chunk.

This chunk is equal to
this chunk -- they're

all equivalent chunks.

There are seven chunks
here, seven chunks here.

So each x must be equal
to two of these objects.

So we get x is equal to, in
this case -- in this case

we had the objects drawn
out where there's two of

them. x is equal to 2.

Now, let's just do a couple
more examples here just so it

really gets in your mind that
we're dealing with an equation,

and any operation that you do
on one side of the equation

you should do to the other.

So let me scroll
down a little bit.

So let's say I have I say
I have 3x is equal to 15.

Now once again, you might be
able to do is in your head.

You're saying this is
saying 3 times some

number is equal to 15.

You could go through your 3
times tables and figure it out.

But if you just wanted to do
this systematically, and it

is good to understand it
systematically, say OK, this

thing on the left is equal
to this thing on the right.

What do I have to do to
this thing on the left

to have just an x there?

Well to have just an x there,
I want to divide it by 3.

And my whole motivation for
doing that is that 3 times

something divided by 3, the 3's
will cancel out and I'm just

going to be left with an x.

Now, 3x was equal to 15.

If I'm dividing the left side
by 3, in order for the equality

to still hold, I also have to
divide the right side by 3.

Now what does that give us?

Well the left hand side, we're
just going to be left with

an x, so it's just
going to be an x.

And then the right hand side,
what is 15 divided by 3?

Well it is just 5.

Now you could also done this
equation in a slightly

different way, although they
are really equivalent.

If I start with 3x is equal to
15, you might say hey, Sal,

instead of dividing by 3, I
could also get rid of this 3, I

could just be left with an x if
I multiply both sides of

this equation by 1/3.

So if I multiply both sides
of this equation by 1/3

that should also work.

You say look, 1/3 of 3 is 1.

When you just multiply this
part right here, 1/3 times

3, that is just 1, 1x.

1x is equal to 15 times
1/3 third is equal to 5.

And 1 times x is the same thing
as just x, so this is the same

thing as x is equal to 5.

And these are actually
equivalent ways of doing it.

If you divide both sides by
3, that is equivalent to

multiplying both sides
of the equation by 1/3.

Now let's do one more and I'm
going to make it a little

bit more complicated.

And I'm going to change the
variable a little bit.

So let's say I have 2y
plus 4y is equal to 18.

Now all of a sudden it's
a little harder to

do it in your head.

We're saying 2 times something
plus 4 times that same

something is going
to be equal to 18.

So it's harder to think
about what number that is.

You could try them.

Say if y was 1, it'd be 2
times 1 plus 4 times 1,

well that doesn't work.

But let's think about how
to do it systematically.

You could keep guessing and
you might eventually get

the answer, but how do you
do this systematically.

Let's visualize it.

So if I have two y's,
what does that mean?

It literally means I have two
y's added to each other.

So it's literally y plus y.

And then to that I'm
adding four y's.

To that I'm heading four y's,
which are literally four

y's added to each other.

So it's y plus y plus y plus y.

And that has got to
be equal to 18.

So that is equal to 18.

Now, how many y's do I have
here on the left hand side?

How many y's do I have?

I have one, two, three,
four, five, six y's.

So you could simplify this
as 6y is equal to 18.

And if you think about it
it makes complete sense.

So this thing right here,
the 2y plus the 4y is 6y.

So 2y plus 4y is 6y,
which makes sense.

If I have 2 apples plus
4 apples, I'm going

to have 6 apples.

If I have 2 y's plus 4 y's
I'm going to have 6 y's.

Now that's going to
be equal to 18.

And now, hopefully, we
understand how to do this.

If I have 6 times something is
equal to 18, if I divide both

sides of this equation by 6,
I'll solve for the something.

So divide the left hand
side by 6, and divide the

right hand side by 6.

And we are left with
y is equal to 3.

And you could try it out.

That's what's cool
about an equation.

You can always check to see
if you got the right answer.

Let's see if that works.

2 times 3 plus 4 times
3 is equal to what?

2 times 3, this
right here is 6.

And then 4 times 3 is 12.

6 plus 12 is, indeed,
equal to 18.

So it works out.