WEBVTT

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

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

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

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

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

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

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

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You could literally go
through the 7 times table.

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

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

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So you would immediately
be able to solve it.

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You would immediately, just
by trying different numbers

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

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

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how to solve this
systematically.

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

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

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just think about it and
do it in your head.

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

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manipulate these equations,
but even more important to

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understand what they
actually represent.

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

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In algebra we don't
write the times there.

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

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to a variable like this, it
just means that you

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are multiplying.

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It's just a shorthand,
a shorthand notation.

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

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

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used in algebra.

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

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

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like xx or times times.

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So in general when you're
dealing with equations,

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especially when one of the
variables is an x, you

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wouldn't use the traditional
multiplication sign.

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

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represent multiplication.

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So you might have 7
times is equal to 14.

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But this is still
a little unusual.

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If you have something
multiplying by a variable

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you'll just write 7x.

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That literally means 7 times x.

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Now, to understand how you can
manipulate this equation to

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solve it, let's visualize this.

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So 7 times x, what is that?

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

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

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So 7 times x.

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So that literally means x
added to itself 7 times.

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That's the definition
of multiplication.

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

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

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So that right there
is literally 7 x's.

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This is 7x right there.

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Let me re-write it down.

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This right here is 7x.

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

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So just saying that
this is equal to 14.

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Let me draw 14 objects here.

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

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

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

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These are equivalent
statements.

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

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

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divide both sides by 7.

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So let me erase
this right here.

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

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

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

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-- a coefficient is just the
number multiplying

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the variable.

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So some number multiplying the
variable or we could call that

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the coefficient times a
variable equal to

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something else.

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

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this case, or divide both
sides by the coefficient.

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

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7 times something divided
by 7 is just going to be

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that original something.

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

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

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

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

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

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This is an equation.

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It's saying that this
is equal to that.

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

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

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to one side and have
it still be equal.

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They were the same thing.

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

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it into seven groups.

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

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four, five, six, seven.

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

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

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to divide the right hand
side into seven groups.

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

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

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

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are going to be equivalent.

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

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

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all equivalent chunks.

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There are seven chunks
here, seven chunks here.

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

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

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

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them. x is equal to 2.

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

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

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

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you should do to the other.

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So let me scroll
down a little bit.

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

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

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You're saying this is
saying 3 times some

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number is equal to 15.

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

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

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is good to understand it
systematically, say OK, this

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

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

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to have just an x there?

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

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And my whole motivation for
doing that is that 3 times

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

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going to be left with an x.

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Now, 3x was equal to 15.

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

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

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Now what does that give us?

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

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

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

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Well it is just 5.

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Now you could also done this
equation in a slightly

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different way, although they
are really equivalent.

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

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

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

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this equation by 1/3.

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

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that should also work.

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

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

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3, that is just 1, 1x.

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

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

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thing as x is equal to 5.

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And these are actually
equivalent ways of doing it.

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

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multiplying both sides
of the equation by 1/3.

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

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bit more complicated.

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

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

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

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do it in your head.

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

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something is going
to be equal to 18.

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

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You could try them.

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

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well that doesn't work.

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

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You could keep guessing and
you might eventually get

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the answer, but how do you
do this systematically.

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Let's visualize it.

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

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

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So it's literally y plus y.

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

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

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y's added to each other.

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

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And that has got to
be equal to 18.

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So that is equal to 18.

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

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How many y's do I have?

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

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

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And if you think about it
it makes complete sense.

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

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

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

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to have 6 apples.

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

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Now that's going to
be equal to 18.

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

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

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

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

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right hand side by 6.

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And we are left with
y is equal to 3.

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And you could try it out.

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That's what's cool
about an equation.

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

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Let's see if that works.

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

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2 times 3, this
right here is 6.

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And then 4 times 3 is 12.

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6 plus 12 is, indeed,
equal to 18.

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So it works out.