When dealing with students who are in math classes in general, I tend to find that the vast majority of them do not understand the basics. By basics, I don’t mean arithmetic, but the real basic inner workings of expressions and equations. In a previous blog post, I broke down the definitions of terms, expressions and equations because so many people do not really know what those three words mean without having to think about it. However, I did not cover how those parts work together, and that’s what I’m going to do here. If you’re like most students, then understanding what I’m going to cover in the following, will eliminate at least half of your problems understanding algebra in general.
Breaking Down Terms
A term is composed of two parts, one of which is optional. The first part is a constant, and a constant is just a number like 5, -6 or 1.2. The second part of a term is the variable portion, and it consists of one or more variables multiplied together. Note that these variables can have exponents as well. While all terms must have a constant portion, they do not have to have a variable portion.
You can add terms if they have the exact same variable portion. For example, you could add 3ab² and 9ab² because their variable portions are the same. You could not add 4ab² and 2a²b. Even though they use the same variables, they are set to different powers, so they are not exactly the same. When two terms have the exact same variable portion, they are called like terms.
To add terms, you make sure that their variable portions are exactly the same, and then you just add the constants. With our example above, we had 3ab² + 9ab². To add these, we just add 3 + 9 to get 12 for the constant portion, and then we just keep the variable portion the same to get 12ab². Subtraction works the exact same way; you could do 5a²b – 8a²b = –3a²b, for example.
It’s worth noting that 5 and -9 are also like terms because they have the same variable portion (ie: they have no variable portion). You add these numbers the same way, obviously, by adding the constants (5 + -9) and leaving the variable portion the same (there is no variable portion) to get -4.
Simplifying an Expression
We know from our previous blog post that an expression is a series of terms added (or subtracted) together. For example, 2a²b + 5ab² – 7ab is an expression. Sometimes you can simplify an expression by adding (or subtracting) like terms.
In the expression 6x² – 4y + 3x², there are two terms that have the exact same variable portion. We can combine them to get 9x² like we learned to do two paragraphs ago. The expression would then become 9x² – 4y. Since there aren’t any terms left that have the same variable portion, we know that the expression is simplified.
Sometimes we’ll need to use the distributive property to simplify an expression. Suppose we have the expression 3(2x – 4) + 9 – 4x. We’ll need to multiply the 3 by the (2x – 4) before we can make any progress. Once we do the multiplication, we’ll be left with 6x – 12 + 9 – 4x. At this point, we can combine like terms as before to get 2x – 3. Again, since we have no like terms, we know that we have simplified the expression as much as possible.