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 algebraic expression.
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 five.
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 minus 𝑐.
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
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
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 𝑧.
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 multiplication expression.
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
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
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 thirty-five.
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.