Hi guys.
Welcome to Algebra 1.
Today's lesson will focus
on the properties of equality.
We're getting close to solving
multi-step linear equations,
and you should understand
these properties
because they allow us
to solve those equations.
Okay, you ready?
Let's go.
Okay.
The equality property
of addition--
or addition property
of equality--
says if we have an equation--
in this case, r equals s--
and r and s
are both real numbers,
then adding t--
another real number--
to both sides
won't change the value
of the equation.
Rather,
it will change the value,
but it will keep it balanced.
If I have r equals s,
r plus t
is equal to s plus t.
Adding the same thing
to both sides
still keeps the equation equal
and balanced.
All right? Okay.
Subtraction property
of equality.
Again, r, s, and t
are real numbers.
We're saying here
that if r equals s,
then r minus t
is equal to s minus t.
So in other words,
subtracting the same value
from both sides
of our equation
does not change
its equality.
The equation
is still balanced,
as long as I subtract
the same thing
from both sides, okay?
Multiplication property
of equality.
Let's think about this.
If the addition property
said adding the same thing
to both sides
didn't change the equality,
the subtraction property said
subtracting the same thing
from both sides
doesn't change the equality.
Then what do you think the
multiplication property means?
I'm sure you guessed it.
If I multiply the same thing
on both sides,
it doesn't change
the equality.
The equation
is still balanced.
So in other words,
if r equals s,
then r times t
is equal to s times t.
As long as I multiply
the same thing
on both sides,
the equality stays intact.
It's still balanced.
Let's keep going.
Division property of equality.
Same general idea:
r, s, and t
are real numbers,
and if r equals s,
then r divided by t
is equal to s divided by t.
As long as I'm dividing
the same thing on each side,
or dividing each side
by the same thing,
the balance of the equation
still stays intact.
It's still equal, all right?
Okay, let's put this stuff
to work here.
Let's look at this equation.
It looks like a lot--
there's more steps
I'll scroll to show you.
We'll work
our way through here.
With these property symbols
on the right,
we figure out
which property allowed us
to transition between
the two steps.
Let's go ahead
and get started.
Step 1: I have 5 times
that quantity, (x plus 2),
equals 30.
Basically they're giving you
the equation right here.
Step 2: Now the left side
is 5x plus 10 equals 30.
Let's throw back to
the properties of real numbers.
Which property let us move
between steps 1 and 2?
Right,
it's the one you've seen
probably since sixth
or seventh grade.
The distributive property
allowed us move
between step 1 and 2.
We'll focus
on the properties of equality.
We're going to keep moving.
From step 2 to step 3,
so now I have 5x plus 10
minus 10 equals 30 minus 10.
Then, step 4:
I have 5x equals 20.
I see the little property
on the right.
So which property let us move
from step 3 to step 4?
If we look at exactly
what happened,
we subtracted 10
on the left side,
and we subtracted 10
on the right side.
We subtracted the same thing
from both sides.
Which property of equality
said that we can subtract
the same real number
from both sides
without changing
the balance of our equation?
Do you remember?
It was
the subtraction property.
Right here I'm going to put
"subtraction."
Good job. Let's keep going.
There's a few more steps.
Let me scroll a little bit.
Six steps here.
We were right here at step 4:
5x equals 20.
Step 5: 5x equals 20.
And they divided each side
of the equation by 5.
Step 6: We have x equals 4.
Let's see what happened.
Between step 5 and step 6,
what happened?
They divided both sides
of the equation
by the same real number--
in this case, 5.
So which property
of real numbers
said that you can divide
both sides of your equation
by the same term
and everything's fine?
It doesn't change
the balance of your equation.
Do you remember?
Division property, right.
So right here,
let's put "division."
That's what allowed us
to move between step 5
and step 6.
So not too bad.
In solving equations
in Pre-Algebra,
you've done this,
without realizing
what you're exactly
you're doing, maybe,
or why
you're able to do that.
You know these steps, how to
move through an equation.
We're giving the properties
that justify why we're able
to make those moves, okay?
All right,
let's try this one.
We'll work our way
through all the steps
of this equation.
When you see
the little property box,
write down
what property allows us
to move
between those two steps.
You've been doing this,
solving equations, all along.
From Pre-Algebra,
you learned what you can do
to both sides of the equation
to work through it.
We're putting
those steps on paper.
We're writing down exactly
what justifies all the moves
we make to solve an equation.
Let's look here. Step 1.
We have negative x minus 4
equals negative 2,
and then in parentheses,
we have x plus 3.
Step 1, they're just
giving you the problem there.
Step 2:
We still have the same thing
on the left--
negative x minus 4.
But the right side,
now it's negative 2x minus 6.
This is a throwback here.
It's a property
of real numbers.
Do you remember
what allowed us
to move between
those two steps?
Distributive property, right.
Good, good, good.
Step 3.
We have negative x minus 4
plus 2x equals
negative 2x
plus 2x minus 6.
Just take that in.
Step 4:
x minus 4 equals negative 6.
I see the little
property box.
Let me set this up
and you can figure out
the property.
In black, it looks like
they're telling me
I added 2x to both sides
of my equation.
Think about which
property of equality
tells you that you can add
the same real number
to both sides
and it doesn't change
the balance of it?
Think about that.
Press pause.
To check your answer,
go ahead
and press play again.
Okay, let's look.
Adding the same thing
to both sides
doesn't change
the balance;
that was
the addition property.
Good--that's the property
that justifies the move
from step 3 to step 4.
All right, let me
scroll down a little bit.
You've got one more to try
for this problem.
Let me get you there.
Step 4.
So we have x minus 4 equals
negative 6.
Step 5: We have x minus 4
plus 4 equals
negative 6 plus 4.
It should look
a little familiar.
Step 6: x equals negative 2.
Let's look here.
What property says that I can
add the same real number
to both sides and it
doesn't change anything?
You got it,
it was another repeat.
It's also
the addition property.
All right!
Great job, guys.
I hope you're feeling
comfortable
with the properties
of equality.
You're ready to solve
multi-step linear equations.
See you soon for that! Bye.
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