When we're dealing with basic arithmetic,
we see the concrete numbers there.
We'll see 23 + 5.
We know what these numbers are right over here
and we can calculate them.
It's going to be 28.
We can say 2 x 7.
We could say 3 divided by 4 (3 / 4).
In all of these cases, we know exactly
what numbers we're dealing with.
As we start entering into the algebratic world –
(And you probably have seen this a little bit already.)
– we start dealing with the idea of variables.
And variables, there are a bunch of ways
you can think about them.
but they're really just values and expressions
where they can change.
The values in those expressions can change.
So for example, if I write
'x + 5.'
this is an expression right over here.
This can take on some value,
depending on what the value of x is.
If x is equal to 1,
then x + 5 – our expression right over here –
Is going to be equal to 1 –
because now x is 1.
It'll be 1 + 5.
So x + 5 will be equal to 6. (x + 5 = 6)
If x is equal to, I don't know, -7, (x = -7)
then x + 5, is going to be equal to –
Well now x is -7.
It's going to be -7 + 5, which is -2.
So notice.
x here is a variable, x here is the variable,
and its value can change depending on the context.
And this is in the context of an expression.
You'll also see that in the context of an equation.
It's actually important to realize the distinction
between an expression and an equation.
An expression is really just a statement of value –
a statement of some type of quantity.
So this is an expression.
An expression would be something like.
well, what we saw over here:
x + 5
The value of this expression will change
depending on what the value of this variable is.
And you could just evaluate it for different values of x
Another expression could be something like ...
I don't know ... y + z.
Now everything is a variable.
If y is 1 and z is 2,
it's going to be 1 + 2.
If y is 0 and z is -1,
it's going to be 0 + (-1).
These can all be evaluated
and they'll essentially give you a value
depending on the values of each of these variables
that make up the expression.
In an equation, you're essentially setting expressions
to be equal to each other.
That's why they're called 'equations.'
You're equating two things.
In an equation, you'll see one expression
being equal to another expression.
So, for example, you could say something like
x + 3 = 1.
And in this situation where you have one equation,
with only one unknown,
you could actually figure out
what x needs to be in this scenario.
And you could possibly even do it in your head.
'What' + 3 is equal to 1? ( __ + 3 = 1?)
Well, you can do that in your head.
Ff I have -2, -2 + 3 is equal to 1. (-2 +3 = 1)
So in this context, an equation is starting to constrain
the value that this variable can take on.
But it doesn't have necessarily constrain as much.
You could have something like:
x + y + z = 5.
Now – this expression is
equal to this other expression.
5 is really just an expression right over here.
And there are some constraints.
If someone tells you what y and z is,
then that constrains what x is.
If someone tells you what x and y are,
then that constrains what z is.
But it depends on what the different things are.
So for example,
if we said y = 3, and z = 2,
then what would x be in that situation?
So if y = 3, and z =2,
then you're going to have –
the left hand expression is going to be
x + 3 + 2 –
which is going to be x + 5 –
This part right over here is going to be 5.
x + 5 = 5
And so what + 5 = 5?
Well now, we're constraining x to be –
x would have to be –
x would have to be equal to 0. (x = 0)
But the important point here –
1) hopefully, you realize the difference
between an expression and an equation.
In an equation, essentially,
you're equating two expressions.
The important thing to take away from here,
is that a variable can take on different values,
depending on the context of the problem.
And to hit the point home,
let‘s just evaluate a bunch of expressions,
when the variables have different values.
So for example, if we had the expression
if we had the expression.
x to the y power,
if x is equal to 5,
and y is equal to 2
y is equal to 2.
then our expression here is going to evaluate to –
Well x is now going to be 5.
x is going to be 5.
y is going to be 2.
it's going to be 5 to the second power.
or it's going to evaluate to
25.
If we change the values –
If we said x –
(Let me do it in that same color.)
If we said x is equal to -2,
and y is equal to 3,
then this expression would evaluate to,
(Let me do in that color.)
– so it would evaluate to -2.
(That's what we're going to substitute for x now,
in this context.)
– and y is now 3 –
-2 to the third power –
which is -2 x -2 x -2,
which is -8.
-2 × -2 = +4.
× -2 again is equal to -8.
is equal to -8
So you see, depending on what the values of these are –
(And we could even do more complex things.)
We could have an expression like
"the square root of x + y and then minus x" ... like that.
If x is equal to – Let's say that x is equal to 1,
and y is equal to 8,
then this expression would evaluate to –
(Well every time we see an x, we want to put a 1 there.)
– so we would have a 1 there.
And you would have a 1 over there.
And every time you would see a y,
you would put an 8 in its place –
– in this context. We're setting these variables to specific numbers.
So you would see an 8.
So under the radical sign, you would have a 1+8 –
so you would have the principal root of 9 – which is 3.
So this whole thing would simplify in this context.
When we set these variables to be these things,
this whole thing would simplify to be 3.
1 + 8 is 9.
Principal root of that is 3.
And then you'd have 3 - 1.
Which is equal to 2.