Simplifying The Expression $5x + 3y + (-x) + 6z$ A Step-by-Step Guide
Hey everyone! Today, we're diving deep into simplifying algebraic expressions. It might sound intimidating, but trust me, it's like piecing together a puzzle. We'll break down the process step-by-step, using a real example to make things crystal clear. So, let's get started and unlock the secrets of simplification!
Understanding the Expression:
Alright, let's take a good look at our expression: . It's a mix of terms with variables (like x, y, and z) and coefficients (the numbers in front of the variables). Our mission, should we choose to accept it, is to make this expression as neat and tidy as possible. Simplifying algebraic expressions is a fundamental concept in mathematics, crucial for solving equations and understanding more advanced topics. The goal is to combine like terms to create a more concise and manageable expression, making it easier to work with and interpret. Simplifying an algebraic expression involves identifying terms with the same variable raised to the same power (like terms) and then combining their coefficients through addition or subtraction. This process reduces the complexity of the expression while maintaining its original value. In our expression, we have terms with x, y, and z. To simplify, we'll focus on grouping the x terms together and then writing the simplified expression. This lays the foundation for solving equations and tackling more complex mathematical problems. Simplifying expressions not only makes them easier to read but also streamlines calculations, reducing the chance of errors. This skill is essential for success in algebra and beyond, allowing you to manipulate equations and solve for unknowns efficiently. By mastering the art of simplification, you'll gain a deeper understanding of algebraic concepts and build confidence in your mathematical abilities. It's like having a superpower that allows you to unravel complex problems with ease and precision. So, let's embark on this journey together, and soon you'll be simplifying expressions like a pro!
Key Principles of Simplification
Before we jump into the nitty-gritty, let's arm ourselves with some key principles of simplification. Think of these as the golden rules that will guide us on our quest. The first golden rule is all about combining like terms. Like terms are those that share the same variable raised to the same power. For example, 5x and -x are like terms because they both have x to the power of 1. But 5x and 3y are not like terms because they have different variables. To combine like terms, we simply add or subtract their coefficients (the numbers in front of the variables). For instance, 5x + (-x) becomes 4x because 5 - 1 = 4. Remember, we only add or subtract the coefficients, not the variables themselves. It's like adding apples to apples, not apples to oranges! Another crucial principle is the commutative property of addition. This fancy term simply means that we can add numbers in any order without changing the result. So, 2 + 3 is the same as 3 + 2. This property is super handy because it allows us to rearrange terms in our expression to group like terms together. This can make the simplification process much easier and more intuitive. The associative property of addition also plays a vital role. This property states that the way we group numbers in addition doesn't affect the sum. For example, (2 + 3) + 4 is the same as 2 + (3 + 4). This allows us to group terms strategically to simplify the process. Another helpful tip is to remember the rules of signed numbers. Adding a negative number is the same as subtracting, and subtracting a negative number is the same as adding. Keeping these rules in mind will help you avoid common mistakes when combining like terms. By mastering these fundamental principles, you'll be well-equipped to tackle any algebraic expression and simplify it with confidence. It's like having a toolbox filled with the right instruments for the job. So, let's keep these principles in our back pocket as we move forward, and we'll see how they work magic in simplifying our expression!
Applying the Facts to Simplify
Now comes the fun part – putting our knowledge into action! Let's take our expression, , and simplify it step-by-step. The first thing we want to do is identify those like terms. Remember, like terms have the same variable raised to the same power. In our expression, we have 5x and -x. These are our like terms, and they're the key to simplifying this expression. Now, let's focus on combining these like terms. We have 5x + (-x). Remember, adding a negative is the same as subtracting, so this is the same as 5x - x. What's 5 - 1? It's 4! So, 5x - x simplifies to 4x. Great! We've tackled the x terms. Now, let's look at the rest of the expression. We have 3y and 6z. These terms don't have any other like terms in the expression, so we'll just leave them as they are. They're happy hanging out on their own. Now, let's put it all together. We simplified 5x + (-x) to 4x, and we still have 3y and 6z. So, our simplified expression is . Ta-da! We've successfully simplified the expression. It's like taking a messy room and organizing it into a neat and tidy space. By applying the principles of combining like terms, we've made the expression much easier to understand and work with. This simplified form is equivalent to the original expression, but it's much more concise and manageable. It's like having a secret code that unlocks the meaning of the expression with just a glance. And that's the power of simplification! By breaking down the expression into smaller, manageable parts and applying the rules of algebra, we've achieved a simplified form that's both elegant and efficient. So, let's celebrate our success and move on to more exciting adventures in the world of algebra!
Fact A: To add like terms, add the coefficients, not the variables.
This statement is absolutely true and crucial for simplifying algebraic expressions. It's the foundation upon which we combine like terms. When we add like terms, we're essentially grouping together quantities of the same variable. The coefficient tells us how many of that variable we have. So, when we add the coefficients, we're adding the number of those variables. The variables themselves act as placeholders, indicating the type of quantity we're dealing with. Think of it like adding apples and oranges. If you have 5 apples and 2 apples, you add the numbers (5 and 2) to get 7 apples. You don't add the word
