Simplifying Algebraic Expressions: A Step-by-Step Guide To $-1(2x+3)-2(x-1)$

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Hey guys! Ever feel like algebraic expressions are just a jumble of letters and numbers? Don't worry, we've all been there. Today, we're going to break down how to simplify the expression βˆ’1(2x+3)βˆ’2(xβˆ’1)-1(2x+3)-2(x-1). It might look intimidating at first, but trust me, it's totally manageable. We'll go through each step together, so you'll be simplifying like a pro in no time. Let's dive in and make algebra a little less scary!

Understanding the Basics of Algebraic Expressions

Before we jump into simplifying our specific expression, let's make sure we're all on the same page with the basic concepts. Algebraic expressions are combinations of variables (like x), constants (like 3), and operations (like +, -, Γ—, Γ·). Simplifying an expression means rewriting it in a simpler form, usually by combining like terms. This makes the expression easier to understand and work with. Think of it like decluttering your room – you're just tidying things up! When we talk about simplifying expressions, we're essentially trying to make them as concise and easy to read as possible. This often involves distributing numbers across parentheses, combining terms that have the same variable, and just generally tidying up the equation. It’s like taking a messy sentence and rewriting it to be clear and to the point. The key here is to maintain the value of the expression while changing its appearance. We're not solving for anything (like finding the value of x); instead, we're just making the expression look neater. This skill is super important because simplified expressions are easier to use in further calculations, whether you're solving equations, graphing functions, or tackling more complex math problems. So, before we get into the nitty-gritty of our example, remember that the goal is clarity and conciseness. We want an expression that's easy on the eyes and ready for action!

The Expression: βˆ’1(2x+3)βˆ’2(xβˆ’1)-1(2x+3)-2(x-1)

Okay, let's take a good look at the expression we're going to tackle: βˆ’1(2x+3)βˆ’2(xβˆ’1)-1(2x+3)-2(x-1). At first glance, it might seem a bit complicated with those parentheses and negative signs. But don't worry, we're going to break it down step by step. This expression consists of two main parts, each involving a set of parentheses: βˆ’1(2x+3)-1(2x+3) and βˆ’2(xβˆ’1)-2(x-1). The numbers outside the parentheses, -1 and -2, are coefficients that we'll need to distribute across the terms inside the parentheses. This is a crucial step in simplifying algebraic expressions. Think of it as unlocking the terms inside the parentheses. We're not just getting rid of the parentheses for the sake of it; we're actually performing a mathematical operation. The distributive property is the key here, which states that a(b + c) = ab + ac. We're essentially multiplying each term inside the parentheses by the term outside. This ensures that we correctly account for the relationships between the numbers and variables in the expression. The negative signs in front of the 1 and 2 are really important too. Remember, a negative sign changes the sign of the terms when we multiply. So, when we distribute -1, we'll be flipping the signs of the terms inside the first set of parentheses. Similarly, distributing -2 will affect the signs in the second set. Keeping track of these signs is crucial to getting the correct simplified expression. So, let's keep this expression in mind, and we'll move on to the first major step: distributing those coefficients.

Step 1: Distributing the Coefficients

The first major step in simplifying our expression, βˆ’1(2x+3)βˆ’2(xβˆ’1)-1(2x+3)-2(x-1), is to distribute the coefficients. Remember, distribution means multiplying the term outside the parentheses by each term inside the parentheses. Let's start with the first part of the expression: βˆ’1(2x+3)-1(2x+3). We need to multiply -1 by both 2x2x and +3+3. When we multiply -1 by 2x2x, we get βˆ’2x-2x. It's pretty straightforward – just change the sign of the term. Next, we multiply -1 by +3, which gives us -3. Again, the sign changes. So, after distributing the -1, the first part of our expression becomes βˆ’2xβˆ’3-2x - 3. Now, let's move on to the second part: βˆ’2(xβˆ’1)-2(x-1). Here, we need to multiply -2 by both x and -1. When we multiply -2 by x, we get βˆ’2x-2x. Simple enough! But here's where we need to be extra careful with the signs. We're multiplying -2 by -1. A negative times a negative gives us a positive, so βˆ’2-2 multiplied by -1 is +2. So, after distributing the -2, the second part of our expression becomes βˆ’2x+2-2x + 2. Now, let's put it all together. After distributing the coefficients, our expression looks like this: βˆ’2xβˆ’3βˆ’2x+2-2x - 3 - 2x + 2. We've successfully gotten rid of the parentheses, and we're one step closer to simplifying the expression. The next step is to identify and combine like terms, which will further streamline our expression.

Step 2: Combining Like Terms

Alright, we've distributed the coefficients and now our expression looks like this: βˆ’2xβˆ’3βˆ’2x+2-2x - 3 - 2x + 2. The next step is to combine like terms. This is where we group together the terms that are similar, making our expression even simpler. So, what are like terms? Like terms are terms that have the same variable raised to the same power. In our expression, we have two types of terms: terms with the variable x and constant terms (numbers without a variable). Let's first identify the terms with x: we have βˆ’2x-2x and βˆ’2x-2x. These are like terms because they both have x raised to the power of 1. To combine them, we simply add their coefficients. So, βˆ’2x+(βˆ’2x)-2x + (-2x) equals βˆ’4x-4x. We've combined the x terms into a single term. Now, let's look at the constant terms: we have -3 and +2. These are also like terms because they're both constants. To combine them, we add them together: βˆ’3+2-3 + 2 equals -1. So, we've combined the constant terms into a single constant. Now, let's put our combined terms together. We have βˆ’4x-4x from combining the x terms and -1 from combining the constants. This gives us the simplified expression βˆ’4xβˆ’1-4x - 1. And that's it! We've successfully combined like terms and simplified our expression. This streamlined version is much easier to work with than our original expression. But before we celebrate, let's just double-check our work to make sure we haven't made any mistakes.

Step 3: Double-Checking Your Work

Okay, guys, we've simplified our expression to βˆ’4xβˆ’1-4x - 1, but before we call it a day, it's always a good idea to double-check your work. Trust me, even the best of us make mistakes sometimes! A quick review can save you from errors and boost your confidence in your answer. There are a couple of ways we can double-check. One way is to go back through each step and make sure we didn't make any arithmetic errors. Did we distribute the coefficients correctly? Did we add the like terms properly? It's easy to miss a negative sign or make a small calculation mistake, so take a moment to review each step. Another helpful technique is to substitute a value for x in both the original expression and the simplified expression. If we get the same result for both, that's a good indication that we've simplified correctly. For example, let's substitute x = 0 into our original expression, βˆ’1(2x+3)βˆ’2(xβˆ’1)-1(2x+3)-2(x-1). When x is 0, the expression becomes βˆ’1(2(0)+3)βˆ’2(0βˆ’1)-1(2(0)+3)-2(0-1), which simplifies to βˆ’1(3)βˆ’2(βˆ’1)-1(3)-2(-1), which further simplifies to βˆ’3+2-3 + 2, which equals -1. Now, let's substitute x = 0 into our simplified expression, βˆ’4xβˆ’1-4x - 1. When x is 0, the expression becomes βˆ’4(0)βˆ’1-4(0) - 1, which simplifies to -1. We got the same result! This gives us a good level of confidence that our simplified expression is correct. If we had gotten different results, we'd know we needed to go back and find our mistake. So, always take the time to double-check your work. It's a crucial step in the problem-solving process!

Conclusion: The Simplified Expression

Alright, guys! We've gone through the entire process of simplifying the algebraic expression βˆ’1(2x+3)βˆ’2(xβˆ’1)-1(2x+3)-2(x-1), and after all our hard work, we've arrived at our final answer. Drumroll, please… The simplified expression is βˆ’4xβˆ’1\bf{-4x - 1}! We started with a somewhat complex expression with parentheses and coefficients, but by carefully distributing the coefficients, combining like terms, and double-checking our work, we've transformed it into a much cleaner and easier-to-understand form. Remember, simplifying algebraic expressions is all about making them as concise and manageable as possible. This not only makes them easier to work with in future calculations but also helps us to better understand the relationships between the variables and constants involved. We've seen how the distributive property and the concept of like terms are key tools in this process. And we've also emphasized the importance of double-checking your work to ensure accuracy. So, the next time you encounter an algebraic expression that looks intimidating, remember the steps we've covered today. Break it down, distribute, combine, and double-check. You've got this! Keep practicing, and you'll become a simplification superstar in no time. And remember, algebra might seem like a challenge, but with the right approach, it's totally conquerable. Great job, everyone! You nailed it!