Simplifying Algebraic Expressions: A Step-by-Step Guide
Hey guys! Today, we're diving into simplifying algebraic expressions, and we're going to break down a specific example: . Don't worry if this looks a bit intimidating at first. By the end of this guide, you'll be a pro at tackling similar problems. We'll go through each step in detail, making sure you understand the logic behind it. So, grab your pencils and paper, and let's get started!
Understanding the Basics of Algebraic Expressions
Before we jump into the problem, let's quickly recap what algebraic expressions are all about. Algebraic expressions are combinations of variables (like x), constants (like 5 or 19), and mathematical operations (like addition, subtraction, multiplication, and division). The goal of simplifying an algebraic expression is to make it as concise and easy to work with as possible, all while maintaining its original value. Think of it like decluttering – we want to get rid of the extra stuff without changing what's fundamentally there.
Key Terms You Should Know
- Variable: A symbol (usually a letter) that represents an unknown value. In our example, x is the variable.
- Constant: A fixed numerical value. In our example, 5, 6, 3, and 19 are constants.
- Coefficient: The number multiplied by a variable. For instance, in the term 5x, 5 is the coefficient.
- Term: A single number, a single variable, or the product of numbers and variables. In our expression, 5x, 6x, 3(-2x), and 3(19) are all terms.
- Like Terms: Terms that have the same variable raised to the same power. For example, 5x and 6x are like terms, but 5x and 5x² are not.
Understanding these terms is crucial because simplifying expressions often involves combining like terms. This is a fundamental concept in algebra, and mastering it will make your life much easier when dealing with more complex problems.
The Order of Operations (PEMDAS/BODMAS)
Before we start simplifying, it's super important to remember the order of operations, often remembered by the acronyms PEMDAS or BODMAS:
- Parentheses (or Brackets)
- Exponents (or Orders)
- Multiplication and Division (from left to right)
- Addition and Subtraction (from left to right)
This order tells us which operations to perform first. In our expression, we'll need to deal with the parentheses first, then any multiplication, and finally, we'll combine like terms through addition.
Step-by-Step Simplification of
Alright, let's dive into simplifying our expression! We'll take it one step at a time, so you can see exactly what's happening. Remember, the goal is to make the expression as simple as possible while keeping its value the same.
Step 1: Distribute the 3
The first thing we need to do is handle the parentheses. We have 3 multiplied by the expression (-2x + 19). This means we need to distribute the 3 to both terms inside the parentheses. This is a crucial step, and it involves multiplying 3 by -2x and then multiplying 3 by 19.
So, let's break it down:
- 3 * (-2x) = -6x
- 3 * 19 = 57
Now, we can rewrite our expression as:
Notice how the 3 has been distributed, and the parentheses are gone! This makes the expression much easier to work with.
Step 2: Identify and Combine Like Terms
Next, we need to identify the like terms in our expression. Remember, like terms are terms that have the same variable raised to the same power. In this case, we have three terms with the variable x: 5x, 6x, and -6x. We also have a constant term: 57.
To combine like terms, we simply add or subtract their coefficients. Think of it like grouping similar objects together. We're going to add the coefficients of the x terms:
5x + 6x - 6x
Now, let's add the coefficients:
5 + 6 - 6 = 5
So, when we combine the x terms, we get 5x. Our expression now looks like this:
Step 3: Write the Simplified Expression
We've done it! We've combined all the like terms, and we're left with a simplified expression. There are no more like terms to combine, and the expression is in its simplest form.
Therefore, the simplified form of is:
That's it! We've successfully simplified the algebraic expression. See, it wasn't so scary after all!
Common Mistakes to Avoid
Simplifying expressions can be tricky, and it's easy to make mistakes if you're not careful. Here are a few common pitfalls to watch out for:
- Forgetting the Order of Operations: This is a big one! Always remember PEMDAS/BODMAS. If you multiply before you distribute, or add before you multiply, you'll likely get the wrong answer.
- Incorrectly Distributing: Make sure you multiply the term outside the parentheses by every term inside the parentheses. Don't forget to pay attention to the signs (positive or negative).
- Combining Unlike Terms: You can only combine terms that have the same variable raised to the same power. Don't try to add 5x and 5x², for example.
- Sign Errors: Pay close attention to positive and negative signs. A small mistake with a sign can throw off the entire answer.
By being aware of these common mistakes, you can avoid them and simplify expressions with confidence.
Practice Problems
Now that you've seen how to simplify one expression, it's time to put your skills to the test! Here are a few practice problems for you to try:
Work through each problem step-by-step, following the same process we used in the example. Remember to distribute, combine like terms, and double-check your work. The answers are provided below, but try to solve them on your own first!
Answers to Practice Problems
How did you do? If you got them all right, congratulations! You're well on your way to mastering simplifying algebraic expressions. If you struggled with any of the problems, don't worry. Go back and review the steps, and try again. Practice makes perfect!
Real-World Applications of Simplifying Expressions
You might be wondering,