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When we're dealing
with basic arithmetic, we see the concrete
numbers there. We'll see 23 plus 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 times 7. We could say 3 divided by 4. In all of these cases, we
know exactly what numbers we're dealing with. As we start entering into
the algebraic world-- and you probably have seen
this a little bit already-- we start dealing with
the ideas of variables. And variables, there's
a bunch of ways you can think about
them, but they're really just values in expressions
where they can change. The values in those
expressions can change. For example, if
I write x plus 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 plus 5, our expression right over here,
is going to be equal to 1. Because now x is 1. It'll be 1 plus 5, so x
plus 5 will be equal to 6. If x is equal to, I don't
know, negative 7, then x plus 5 is going to be equal to--
well, now x is negative 7. It's going to be negative 7
plus 5, which is negative 2. So notice x here is a
variable, and its value can change depending on the context. And this is in the
context of an expression. You'll also see it 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 what we saw over here, x plus 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 plus z. Now everything is a variable. If y is 1 and z is 2,
it's going to be 1 plus 2. If y is 0 and z is
negative 1, it's going to be 0 plus negative 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. 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. An equation, you'll
see one expression being equal to
another expression. For example, you could say
something like x plus 3 is equal to 1. And in this situation where you
have one equation with only one unknown, you can
actually figure out what x needs to be
in this scenario. And you might even
do it in your head. What plus 3 is equal to 1? Well, you could do
that in your head. If I have negative 2,
plus 3 is equal to 1. In this context, an
equation is starting to constrain what value
this variable can take on. But it doesn't have to
necessarily constrain it as much. You could have something like
x plus y plus z is equal to 5. Now you have 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 you're going to get an x. If someone tells
you what x and y is, then that constrains what z is. But it depends on what
the different things are. For example, if we said y is
equal to 3 and z is equal to 2, then what would be
x in that situation? If y is equal to 3
and z is equal to 2, then you're going to have
the left-hand expression is going to be x plus 3 plus 2. It's going to be x plus 5. This part right over here is
just going to be 5. x plus 5 is equal to 5. And so what plus
5 is equal to 5? Well, now we're
constraining that x would have to be equal to 0. Hopefully you realize
the difference between expression and 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. For example, if we had the
expression x to the y power, if x is equal to 5
and y is equal to 2, then our expression here
is going to evaluate to, well, x is now 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 changed the values-- let
me do that in that same color-- if we said x is equal to
negative 2 and y is equal to 3, then this expression
would evaluate to-- let me do it in
that-- negative 2. That's what we're
going to substitute for x now in this context. And y is now 3, negative
2 to the third power, which is negative 2 times
negative 2 times negative 2, which is negative 8. Negative 2 times negative 2 is
positive 4, times negative 2 again is equal to negative 8. We could do even
more complex things. We could have an expression
like the square root of x plus y and then minus x, like that. 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'd have a 1 over there. And every time you see a y,
you would put an 8 in its place in this context. We're setting these variables. So you'd see an 8. Under the radical sign,
you would have a 1 plus 8. So you'd have the principal
root of 9, which is 3. 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 plus 8 is 9. The principal root of that is 3. And then you'd have 3 minus
1, which is equal to 2.