Hey guys,
welcome to Algebra 1.
We're reaching that point
where it's time
to solve
multi-step linear equations,
but first,
I want to make sure
you understand
the properties of real numbers.
You've seen some
of this in Pre-Algebra,
but some may be new to you.
You ready? Let's go.
Okay.
Before talking about
the properties of real numbers,
you should know
what a real number is.
The definition
of a real number--
it really can be
a small number, say 1.
It can be a large number,
2,000.
It can be negative, -5.
It can be positive, 4.
It can be a fraction.
It can be a decimal.
It can be rational,
which means it has a decimal
that will terminate, say 2.5.
Or it could be irrational--
pi for example.
Goes on and on forever,
never repeats, never ends.
Basically a real number
is everything that's not
an imaginary number.
I know that sounds funny,
but there are
imaginary numbers.
You'll see them
when you get to Algebra 2.
But a real number
is all of the numbers
that we'll deal with
in this course.
Real numbers have properties
you must know.
One that we've used
is the distributive property,
which says,
if I have three real numbers--
we'll call them
r, s, and t;
variables we know
we can replace
with any value
that we want to.
If I were given
this setup,
where I have r multiplied
by the quantity (s+t),
I know that I need to multiply r
times the first term, so rs,
and add that to the product of r
times the second term, rt.
We've handled that before.
That's one of the ones
that isn't new to you.
Another one's
the associative property.
The associative property
tells me
that if I'm adding numbers--
here, the associative property
of addition--
I can group them together
in any way I want
when adding.
For example, if I were adding
these three terms, r, s, and t,
and I wanted to first add
r and s, and then add t,
I'd get exactly
the same answer
if I added s and t together
first, and then added r.
Let me show you an example.
Let's say r is 1,
s is 2, t is 3.
Let's just use 1, 2, 3.
On the left side,
I'm going to start out
by adding
r and s together.
On the right,
I'll add s and t first.
Is it the same answer?
Let's get more room
to work up here.
So, okay, on the left,
1+2, I know that's 3.
Then 3+3, I know that's 6.
On the right, it's telling me
first add the 2+3.
So 2+3, I know that's 5.
1+5 is also 6.
That's
the associative property.
You can group them together
however you want when adding.
It doesn't change
the value on either side.
Now multiplication,
basically the same idea
except this time
I'm multiplying.
If I wanted to multiply
r and s together
and then multiply
that product times t,
I'd get the exact same answer
if I multiplied
s and t together first,
and then multiplied that
times r.
Let me show you
this one.
Let's get work
out of the way here
so we have some space.
Let's say again r, s, and t
are 1, 2, and 3.
Here we have 1x2x3.
1x2x3.
Let's say on the left
I multiply
1 and 2 together first,
and on the right
I multiply
2 and 3 together first.
1x2, that's 2.
Then 2x3, that's 6.
Now on the right side--
just to keep it separate,
let's scoot that away.
We have some space here.
On the right side,
2x3, that's 6.
Then 1x6 is also 6.
That's the associative property
of multiplication,
which tells you
you can group together terms
when multiplying
however you want,
in whatever order you want,
and it doesn't change anything.
All right,
let's keep moving.
Commutative property.
What that tells you
is for addition
you can add terms together
in whatever order you want,
it doesn't change anything.
And then for multiplying,
you can multiply
in whatever order
and it doesn't
change anything.
Let me show you one.
Let's say this time
r is 1, s is 2.
That's telling us 1+2
is the same as 2+1,
and that's true, right?
1+2 is 3.
2+1 is 3.
Changing the order of addition
doesn't change the answer.
Then, like I said, for
multiplication, same idea.
Let's say now
it's 1x2 and 2x1.
1x2, that's 2.
2x1, that's also 2.
Changing the order
of multiplication
doesn't change your answer.
Let's keep going.
Identity.
This one may be new,
but you may have seen it
in Pre-Algebra.
The identity property
of addition tells me--
I see right here r+0 = r.
Adding 0 to a term
doesn't change its value.
We know that intuitively,
but this is putting it
on paper mathematically.
For example,
let's say this time r is 4.
4+0 is 4.
Adding 0 to anything
doesn't change its value.
Now for multiplication.
Let's think about this
for a second.
What could you multiply
a number by
and not change
its value at all?
You got it? One.
Multiplying a number by 1
doesn't change the value
of that number.
That's the identity property
of multiplication;
it says
any real number times 1
is just that real number.
That's the answer
I get back.
For example,
4x1 is still 4.
That's the identity property
of multiplication.
Let's keep moving.
Inverse.
The inverse property
of addition
tells me
is if I have a real number
and I add its opposite to it,
the answer is 0.
For example, let's say
r is 4--still 4.
4+(-4),
so 4 plus its inverse, is 0.
That's the inverse property
of addition.
Adding the inverse,
adding a number's inverse
to it, is 0.
All right, now, multiplication
is a little different.
The inverse property
of multiplication
tells you if you multiply
a number by its reciprocal,
the answer is 1.
This is what I mean.
Let's say rx 1/r = 1.
I'm going to use
some numbers here
to prove this to you,
so you'll see.
Let's get this out of our way,
get rid of that.
Let's slide that a little bit,
get some space.
Let's say, again,
r is 4.
Multiplying 4
by its reciprocal, so by 1/4,
that answer is going to be 1.
Let me show you.
Remember earlier we said
we could make
a whole number a fraction by
giving it a denominator of 1?
Doesn't change its value
and it helps us
work with fractions.
I'll look at this
like a multiplication problem
of 4/1 x 1/4.
I'll multiply straight across
to get this answer.
4x1, that's 4.
Denominators, 1x4, that's 4.
For my final answer
4 divided by 4,
I know that answer is 1.
Any number
times its reciprocal,
its answer is 1.
That's the inverse property
of multiplication.
All right, doing good.
Let's keep going.
Closure.
This one may be new.
What closure tells us,
the closure property
of addition,
is if I add two real numbers
together,
my answer is
a real number.
I'll never add
two real numbers together
and get an imaginary number.
For example,
let's say r is 4, s is 5.
I know 4 is a real number
and 5 is a real number.
When I add those together,
I get 9.
9 is also a real number.
That's all that means,
that's all that means.
Closure property
of multiplication.
That says if I multiply
two real numbers together,
the answer is also real.
Here we go.
Let's say r is 4
and s is 5.
4x5.
4 is a real number
and 5 is a real number.
The product of 4 and 5 is 20.
20 is also a real number;
that's all that means.
When you multiply
two real numbers together,
the answer is also
a real number.
All right, now let's practice
a little bit.
We have a set
of equations here,
two equations.
They're asking which property
justifies the work?
I know it looks like a lot--
there's a lot of
terms and variables.
Take a deep breath,
look at it,
and we'll break it down to
figure out what's going on.
If I look
at my first equation,
I see I have 2x+3
on the left,
and on the right
I have (4x+1)
and I'm adding x to that.
Second equation, I still have
2x+3 on the left,
so the left sides of these
equations are the same.
But the right side,
something's changed.
You see here that they
switched the order?
Now (4x+1),
it's still in parentheses,
but now it's
the second group listed;
it's listed
towards the back.
That x term,
that's listed up at the front.
I'm still adding things
together,
but they've changed
the order of addition.
Think back.
Which property
of real numbers said
you can change
the order when adding
without changing anything?
Remember?
It's one
you've seen in Pre-Algebra.
It's
the commutative property...
of addition.
That's exactly
what's going on here.
You're changing the order
of what you're doing--
changing the order
of your addition,
but it's not changing
the answer for these.
All right, good job,
let's keep moving.
Example 2.
This one's a little different.
Which property
justifies the work
between step 2 and step 3?
We've got to look at this
in parts, individually.
Look at what's
going on from step to step
to figure out
which property
we'll need
to answer this question.
Let's start
breaking this down.
Step 1 starts out
by giving me the equation.
X+4 = 10.
Step 2, things have changed
a little bit.
I have x+4 on the left,
but they added a -4.
I have that 10
on the right,
but they've added a -4.
File that in your memory,
keep ahold of that.
Then I see step 3.
I have x+0.
It seems like what
happened here, 4+(-4),
that's where the 0 came from.
Then I have 10+(-4),
which is 6.
I'm going to stop there
because this is what
I'll focus on.
Here I have 4
and I added a -4.
It's really the left side,
is the action
to watch here.
If you had 4, added -4,
and your answer was 0,
do you remember
what property said
that if you add a number
and its opposite,
the answer was 0?
That one was
the inverse property...
of addition.
That is the property
that justifies the work
from step 2 to step 3.
I have 4.
I added its inverse
and then the answer was 0.
That's what allowed me
to move from step 2 to step 3.
All right, doing good.
All right.
It's time for you to try.
Figure out which property
justifies the work
between step 3 and step 4.
I'll set you up
and let you go.
I want you to pause,
see how you do,
then press play
to hear the answer.
I see for step 3,
we had x+0 = 6.
Step 4, I have x = 6.
It looks like
what happened here,
the action was
on the left side.
Figure out which property
tells you
that adding 0
to a real number
doesn't change the value
of that real number.
Press pause,
think about that,
and then press play when
you want to compare answers.
Let's see how you did.
Adding 0 to a real number
doesn't change the value
of that real number.
That is the identity property
of addition.
That property
allowed us to move
from step 3 to step 4.
Adding 0 to a real number
doesn't change its value.
All right, good job!
Let's keep going.
We have more properties
of real numbers.
Then I'll let you practice
again for a bit.
One property
is the reflexive property,
which states
that any real number
is equal to itself, right?
We already know that,
really.
For example, 4 = 4.
That's true.
That's all the reflexive
property means--
any real number is equal
to itself, and that's it.
Keep moving.
Multiplicative property of 0.
What this property
tells me
is that the product of any
real number and 0
is always 0.
That's what that means.
Let's say rx0 is 0.
That's the multiplicative
property of 0.
Multiplying a real number
times 0 is 0.
Keep moving,
got a couple more.
The zero product property.
Similar to the multiplicative
property of 0,
but a little different.
The zero product property--
say that three times fast--
the zero product property
states that
if I have a product--
if I multiply
two real numbers together,
here, r and s--
and the answer is 0,
that means that
either r was 0, or s was 0.
That's all the zero product
property means--
if you multiply
two real numbers together
and your answer was 0,
one of those real numbers
had to be 0.
For example,
let's say r is 4 again.
It's the favorite
number today.
For me to multiply
something by r
and get 0,
I must have used 0.
That's all that means.
You have a product
of two numbers.
Your answer is 0.
One number has to be 0.
That's all that means.
All right,
moving right along.
Symmetric property.
What that means here,
if r = s, then s = r.
That's all that means.
Symmetric property.
You can flip around
your equation
without changing
the value at all.
Here,
I'm going to use a variable
to represent
a real number.
I'm going to say if x = 4
then 4 = x.
That's all that means.
Those two real numbers
have the same value.
The equation can be
in either order that I want.
It doesn't change
its meaning at all.
For example, x = 4 is the exact
same thing as 4 = x.
All right, doing good.
Transitive property.
This is one of my favorites.
I like puzzles,
and it reminds me
of figuring out
a puzzle here.
What the transitive property
means is if r = s,
and s = t, then r = t.
Pause,
think about that for a second.
If r = s, and s = t,
then r = t.
Let me show you
what I mean.
Let's say, if r = s...
okay, we'll take that,
we'll believe that.
Now they're saying s = t.
So that means,
basically, this:
r = s, okay,
but now they're saying s = t,
so I'm going to take the s away
and replace it with t.
Let's make that look
like a t here.
All right,
replace that with t.
That's telling me
that r must equal t.
That's
the transitive property.
You may hear it
called "the chain rule."
It means the same thing.
It's like cutting out
the middle man.
If r = s, and s = t,
let's just cut out s,
cut out the middle, r = t.
All right,
doing good.
Got an example.
Which property
justifies the work?
Remember the steps we took
when we were solving these
a minute ago.
Step back from it.
Take a deep breath,
then get ready to attack it.
It looks like
on the first equation,
I have 5x+3 = 2x+9.
Second equation
I have 2x+9 = 5x+3.
Okay, so what actually
happened here?
What's the difference
between these two equations?
They just swapped the order
that they wrote it.
First the 5x+3 came first,
then the 2x+9 was second.
Then they just reversed
the order.
Which property told me
when working with an equation
that I can swap the sides
of my equation?
You got it--that was
the symmetric property.
That's what they're
showing you right here,
that you could swap
the left and right sides
without changing the meaning
of your equation.
All right,
let's keep going.
"Use the zero product property
to explain why the solution
to 5x = 0 is 0."
This is one
I want you to try.
Look back and see what that
zero product property said,
and then look
at this equation, this 5x = 0,
and tell me why the answer
has to be 0.
Use the zero product property
to explain it to me.
Take a minute.
When explaining things
in math,
you mix words
and numbers.
You need
to show me a couple things,
then write out
a couple things.
Take a stab at it.
Then press play when
you're ready to compare.
Let's see how you did.
I'll show you
how I thought through it.
If your thinking process
was similar to mine--
we won't think
about it the same way--
but if we have similar ideas,
you got it right.
The zero product property
says that if I multiply
two numbers together--
two real numbers together--
and the answer was 0,
then one of the numbers
had to be 0, right?
I'll write that
down here.
Zero product property.
Remember that that said
if rs = 0,
so if the product
of two real numbers was 0...
then r = 0 or s = 0.
One of these numbers
had to be 0
if the product
of those numbers is 0.
Then, looking at the equation
they gave us, this 5x = 0,
if the product of
these two real numbers is 0,
then one of these numbers
must be 0.
Let's split them up
to consider them separately.
This is really the product
of 5 and x, right?
5, that's kind of like our r.
Let's get
a few things out of the way
to really see this clearly.
Let's move that up
a bit.
Move that over.
We do want
the multiplication too.
All right,
here we go.
Zero product property:
rs = 0.
I'll write that
right above here
when I get the pen.
All right, rs = 0.
All right,
zero product property
says if rs is 0,
then either r is 0
or s is 0.
Let's look at this.
5 is like our r here,
right?
If r is 5 then I know
it can't be 0, right?
5 does not equal 0.
S is our unknown here--
and s, in this case, is x--
so x is going
to have to have a value of 0.
Because the only way that I
can multiply 5 times a number
and that number be 0--
and that product be 0--
is if that number was 0,
the number
that I multiplied 5 by.
X must be 0
because r = 5 in this case.
The first real number is 5;
they gave us that.
That second real number
must be 0
if that product is 0.
All right.
Okay, guys, you've reached
the end of reviewing
your properties
of real numbers.
I hope you understand
of what we reviewed here.
It will come in handy with
multi-step linear equations.
See you soon.
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