Let’s take a look at how we
evaluate simple algebraic expressions. Algebra is the branch of
mathematics that deals with variables. And a variable is a symbol that
represents an unknown quantity. But those are all just words. Let’s look at some examples.
𝑛 plus three. In this expression the letter 𝑛 is
the variable. This 𝑛 represents an unknown
quality, an amount that we don’t know. And 𝑛 plus three together is an
algebraic expression. It’s algebraic or from algebra
because this expression 𝑛 plus three contains a variable; it contains 𝑛.
And it’s an expression because it
contains a number or numbers and at least one operation. So here we have the operation of
addition and the number three to make this an expression. 𝑛 plus three is an example of an
But remember our goal is solving
algebraic expressions. So in order to do that, we’re going
to need a little bit more information. Here is our extra information. Evaluate 𝑛 plus three if 𝑛 equals
My first step here is just to copy
down exactly the expression 𝑛 plus three. Our next step is to replace the 𝑛
with a five. After that I add five and
three. And I understand that 𝑛 plus three
when 𝑛 equals five is eight. So we’ve evaluated or solved our
expression 𝑛 plus three with the given information.
Let’s take a look at this
example. Solve the expression below if 𝑐
equals seven and 𝑎 equals five. Our expression is eight plus 𝑎
We just need to start by copying
down the expression exactly how it’s listed in the problem. Then I wanna replace 𝑎 and 𝑐 with
their corresponding values, in this case 𝑎 being equal to five and 𝑐 being equal
to seven. Now I have an expression that’s
full of numbers and I can follow the order of operations. I’ll add 𝑎 and five to give me
thirteen. I added the 𝑎 and five first
because in the order of operations we want to add and subtract from left to
right. And finally we’ll subtract the
seven from the thirteen which equals six.
When we’re given these values for
𝑎 and 𝑐 — when we’re given 𝑐 equals seven and 𝑎 equals five — we can understand
that eight plus 𝑎 minus 𝑐 equals six.
Remember how I said earlier that
algebraic expressions contain at least one operation. So far we’ve only seen examples of
addition and subtraction in expressions. But expressions can also contain
multiplication and division.
In fact here’s an example of an
algebraic expression with multiplication. Are you curious how this is an
example of an expression with multiplication? Five 𝑑 equals five times 𝑑. In algebra the multiplication sign
is often omitted. You might see something like nine
𝑠, three 𝑞𝑟, or even 𝑧𝑦. Nine 𝑠 is the same thing as saying
nine times 𝑠, three 𝑞𝑟 equals three times 𝑞 times 𝑟, and 𝑦𝑧 equals 𝑦 times
Now I want you to take a look at
the numbers that I’ve highlighted in green. In algebra there’s a special name
for these numbers that are being multiplied by variables. These numbers are called “the
coefficient.” Coefficient is a factor of a
Here’s an example of a
multiplication expression. Evaluate seven 𝑤 if 𝑤 equals
four. First copy down the expression. Next I’m gonna replace my 𝑤 with
the four. And I also added a multiplication
symbol in this time. After that I multiply seven times
four. The solution to this expression is
seven times four, which is twenty-eight.
Here’s a slightly harder
example. Evaluate 𝑦 squared minus four plus
three if 𝑦 equals six. Even though this expression has
three different operations, we always start with the same procedure. I’m sure you guessed “copy down the
expression.” And if you did, you would be right;
that’s the first thing we need to do. Next we’re gonna replace 𝑦 with
six because that was our given value.
As you solve more and more
complicated expressions, the most important thing you can remember is that you have
to follow the order of operations. We have correctly substituted the
six for the 𝑦. But now what operation comes
first? Since we don’t have any parentheses
or brackets in this problem, order of operations would tell us to go ahead and solve
your exponents next. So I’ve solved for six squared
which equals thirty-six and copied the rest of the problem down.
Following order of operations, I
now need to add and subtract from left to right. In this step, I subtracted four
from thirty-six. And our next order of operations
step will be to add thirty-two plus three. Our final answer for this
expression when 𝑦 equals six, 𝑦 squared minus four plus three equals
Okay, so let’s review. The first thing that we’ll do when
we’re trying to evaluate algebraic expressions is to substitute given values for the
variables. And after that you need to be very
careful to follow the order of operations to evaluate each operation that’s in your
expression. The best way for you to get good at
evaluating expressions is to try it and then practice. So now it’s your turn.