Simplifying Expressions: A Step-by-Step Guide
Hey everyone! Today, we're diving into the world of algebraic expressions and learning how to simplify them like pros. We'll be tackling the expression 5(x+2) - (3x - 6). Don't worry if it looks a bit intimidating at first; by the end of this, you'll be simplifying with ease! This is a fundamental skill in mathematics, so let's get started. Simplifying expressions is a core concept, and once you grasp it, you’ll find that many other math topics become much easier to understand. The key here is to break down the problem into smaller, manageable steps. We'll cover everything from the distributive property to combining like terms, and you'll soon be simplifying expressions with confidence. So, grab your pencils and let's jump right in. Let's make this fun, guys!
Before we begin, remember that an expression is a mathematical phrase that contains numbers, variables, and operations, but no equal sign. Our goal is to rewrite the given expression in a simpler form without changing its value. This is similar to simplifying a fraction – you're changing how it looks, but not what it represents. This process involves several key steps. First, we'll apply the distributive property to remove any parentheses. Then, we will combine like terms to further simplify the expression. Finally, we'll present our simplified expression, which will be equivalent to the original expression but in its most reduced form. Ready? Let's roll!
Step 1: Distributing the 5
Alright, the first step in simplifying the expression 5(x + 2) - (3x - 6) is to deal with those pesky parentheses. We have two sets of parentheses to deal with, and we'll start with the first one: 5(x + 2). Here, the 5 is multiplied by everything inside the parentheses. This is where the distributive property comes into play. The distributive property states that a(b + c) = ab + ac. In our case, a = 5, b = x, and c = 2. This means we need to multiply 5 by both x and 2.
So, 5 * x = 5x and 5 * 2 = 10. Therefore, when we distribute the 5, 5(x + 2) becomes 5x + 10. Remember, this is like saying we have five groups of (x + 2). Each group has an x and a 2, so overall we have five xs and five 2s, which is ten.
Great job! Now our expression looks like this: 5x + 10 - (3x - 6). Keep in mind that we're only working on the first part of the original expression. Next, we will deal with the second set of parentheses. This might be a little trickier, but don't sweat it; you've got this! Understanding the distributive property is crucial for a wide range of algebraic manipulations. It lets you rewrite expressions in a more convenient form, making it easier to solve equations and analyze mathematical relationships. Take your time, focus on each step, and you'll master this technique in no time. Think of it like a secret code: once you break the code, you unlock the ability to simplify a vast variety of expressions.
Now we're moving onto the second part, which can sometimes be tricky. The minus sign in front of the second set of parentheses is like multiplying the entire expression (3x - 6) by -1. Because of this, it can cause errors if not handled with care. To simplify -(3x - 6), we distribute the negative sign to both terms inside the parentheses.
Step 2: Distributing the Negative Sign
Now, let's tackle the second part of our expression, - (3x - 6). Notice that there is a negative sign in front of the parentheses. This is equivalent to multiplying the entire expression inside the parentheses by -1. So, we'll distribute that negative sign. Remember, when you multiply a term by -1, you change its sign. This is the distributive property again, but this time with a negative twist. Therefore, -1 * 3x = -3x and -1 * -6 = +6. When multiplying a negative number by a negative number, the result is positive. Be careful with this step; it's a common area where mistakes can happen. So, - (3x - 6) becomes -3x + 6. Now, our expression looks like this: 5x + 10 - 3x + 6. Awesome! We're almost there. The key is to pay close attention to the signs – the positives and the negatives – because they're critical. Remember, the negative sign in front of a parenthesis changes the sign of each term inside when you remove the parenthesis. This often causes confusion, so make sure you're careful when working through these steps.
We have now expanded both sets of parentheses by applying the distributive property. Next, we'll focus on combining like terms.
Step 3: Combining Like Terms
Alright, now that we've distributed the 5 and the negative sign, we have the expression 5x + 10 - 3x + 6. Our next step is to combine like terms. Like terms are terms that have the same variable raised to the same power. In this case, we have two types of terms: terms with x (the variables) and constant terms (numbers without a variable). So, we can combine 5x and -3x. This is done by subtracting the coefficients of the variables. That is, 5 - 3 = 2, so 5x - 3x = 2x. In addition, we also have the constant terms 10 and 6. Combining these, we get 10 + 6 = 16. Therefore, we can rewrite the expression as 2x + 16. Remember, only like terms can be combined. You can't combine a term with x and a constant term.
Combining like terms simplifies the expression by grouping similar elements. This makes the expression more concise and easier to interpret. It's like organizing your desk: you put all your pens together, your papers together, etc. Similarly, you put all the 'x' terms together and the constant numbers together.
We've simplified the expression considerably by combining like terms. Let's see how our steps look in a nutshell: first, we distributed the number outside the parentheses, and then we distributed the negative sign. Finally, we combined like terms. With each step, the expression became simpler and easier to manage. Now, let’s see the solution.
Step 4: The Simplified Expression
Great job, everyone! We have successfully simplified the expression. Our original expression was 5(x + 2) - (3x - 6). After distributing the 5, we got 5x + 10 - (3x - 6). Then, distributing the negative sign, we obtained 5x + 10 - 3x + 6. Finally, combining like terms, we arrived at 2x + 16. Therefore, the simplified form of 5(x + 2) - (3x - 6) is 2x + 16.
This is the simplest form of the expression. There are no more parentheses to remove, and all like terms have been combined. The expression 2x + 16 is equivalent to the original expression 5(x + 2) - (3x - 6), meaning that if you substitute any value for x in both expressions, you'll get the same result. The simplification of the expression is now complete. Great work, everyone! You've successfully navigated the steps of simplifying the expression.
Conclusion
So there you have it, guys! We've successfully simplified the expression 5(x+2) - (3x - 6). We learned the importance of the distributive property and how to combine like terms to achieve a simpler form. Remember that practice makes perfect. Try working through similar examples on your own. Simplifying expressions is a fundamental concept in algebra and is essential for more advanced topics. I hope this guide was helpful. Keep practicing and you’ll master it in no time! Keep up the great work! If you have any questions, feel free to ask. Cheers!