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. An 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 for 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 eight and five to give me 13. 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 13, 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 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 28. 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 36 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 36. And our next order of operations step will be to add 32 plus three. Our final answer for this expression when 𝑦 equals six, 𝑦 squared minus four plus three equals 35.
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.