Chapter 1, Lesson 6
In this lesson students will:
- understand the difference between an algebraic expression and an algebraic equation
- simplify an algebraic expression
- expand or factor an algebraic expression
- solve an algebraic equation involving one variable
Algebraic Expressions & Equations
An algebraic expression is a string of terms separated by addition or subtraction. Within algebraic expressions can be found letters that represent unknown values. These letters are called variables. A term of an algebraic expression involves a real number, a variable, or both. When both real numbers and variables are side-by-side, it is understood that they are connected by the multiplication operation.
Let’s consider the expression . Here, , and 3 are the terms of this algebraic expression. The letters and are used to represent unknown values, so we call these letters the variables in this expression. Notice that means 2 times .
An algebraic equation contains two algebraic expressions that are connected using an equal sign. For instance, would be considered an algebraic equation, with and being the two expressions connected by an equal sign.
In general, we often simplify an algebraic expression, and we often solve an algebraic equation. To simplify an algebraic expression, we follow the order of operations agreement when we can and try to collect like terms. To solve an algebraic expression, we try to isolate one variable at a time, if possible.
Expanding and Collecting Like Terms
To collect like terms means to find terms in our expression that share not only the same variable letter, but also the same power of that variable. For example, and would be considered like terms since the power of the variable is one in both cases. Notice that, and would not be like terms since the powers of the variable are different.
When working with algebraic expressions, we will often have to expand before we are able to collect like terms. Consider the expression . The order of operations agreement states we should look to work inside the brackets first (P-step). Unfortunately, there are no terms that we can add together. Since we are unable to do any work inside the brackets, and there are no exponents to worry about (E-step), we move to the next step which is multiplication (M-step). While there is multiplication involved in the terms and , we cannot really do this multiplication since we do not know what is as a number. Notice that there is one other understood multiplication between the 3 and the ; since the number 3 lies beside a bracket, the operation here is multiplication. Expansion in this case involves distributing the 3 to both the and the 1. The video below will review this kind of expansion, and will work through a few more advanced examples.
Finding a Common Factor
In our course, we may come across a few instances in which it would be helpful to find a common factor. Two terms that share a similar variable will have a common factor. Also, if the numerical portions of each term share a factor, it will be a part of the common factor as well. In general, when finding a common factor, we (1) find the greatest common factor of the numerical portions of our terms, and (2) find the smallest power of the variable portions of our terms. This process will be easier explained through the following video.
Isolating One Variable
Now that we have a better understanding of how to work with algebraic expressions, let’s turn our focus to working with algebraic equations. We will begin by working with a single variable. The next couple of chapters deal with situations involving more than one variable.
Equations can be a little daunting since it can sometimes be unclear on where to start. Let’s try to standardize our approach to working with equations involving a single variable by:
- Removing brackets using the expansion technique discussed earlier.
- Simplifying the algebraic expressions on each side of the equal sign by collecting like terms, if possible.
- Collecting all terms involving variables on the left-hand side of the equal sign.
- Collecting all terms not involving variables on the right-hand side of the equal sign.
The video below will guide you through a few examples of the four-step process.