Simplifying Algebraic Expressions What Is (1.6x + 7.3) + (-0.6x + 2) - (7.8 - 3.4x)
Hey everyone! Let's break down how to simplify the algebraic expression (1.6x + 7.3) + (-0.6x + 2) - (7.8 - 3.4x). This might look a bit intimidating at first, but don't worry, we'll go through it step by step. It's all about combining like terms and getting rid of those parentheses in the right order. So, grab your pencils (or keyboards!) and let's dive in!
Understanding the Expression
Before we start crunching numbers, let's take a good look at what we're dealing with. The expression (1.6x + 7.3) + (-0.6x + 2) - (7.8 - 3.4x) is made up of several terms. We've got terms with the variable 'x' (like 1.6x and -0.6x), and we've got constant terms (just plain numbers, like 7.3 and 2). The goal here is to simplify this expression by combining these like terms. That means adding or subtracting the 'x' terms together and adding or subtracting the constant terms together. But before we can do that, we need to handle those parentheses. Remember, the order of operations (PEMDAS/BODMAS) tells us that we need to deal with parentheses first. This involves distributing any negative signs that are lurking outside the parentheses, ready to change the signs of the terms inside.
Our first step in simplifying the expression involves addressing the parentheses. When we have parentheses in an algebraic expression, it's crucial to handle them correctly to avoid making mistakes. Parentheses act like little containers that group terms together, and we need to unpack them before we can combine like terms across the entire expression. In our case, we have three sets of parentheses: (1.6x + 7.3), (-0.6x + 2), and (7.8 - 3.4x). The first set, (1.6x + 7.3), is being added to the rest of the expression. This is pretty straightforward – we can simply remove these parentheses without changing any signs, since there's an implied positive sign in front of them. The second set, (-0.6x + 2), is also being added, so we can remove these parentheses as well without any sign changes. However, the third set, (7.8 - 3.4x), is being subtracted. This is where we need to be careful! The negative sign in front of these parentheses means we need to distribute that negative sign to each term inside the parentheses. This is like multiplying each term inside by -1. So, the +7.8 becomes -7.8, and the -3.4x becomes +3.4x. Once we've correctly handled the parentheses, the expression will be ready for the next step: combining like terms. This careful approach ensures that we maintain the integrity of the expression and arrive at the correct simplified form.
Distributing the Negative Sign
This is a crucial step, so let's make sure we've got it down. When we see a minus sign directly in front of a set of parentheses, like in -(7.8 - 3.4x), it's like we're multiplying the entire contents of the parentheses by -1. Think of it as a little sign-changing ninja! This means that every term inside those parentheses has its sign flipped. So, the positive 7.8 becomes a negative 7.8, and the negative 3.4x becomes a positive 3.4x. It's all about flipping those signs! Why is this so important? Well, if we don't distribute the negative sign correctly, we'll end up with the wrong answer. It's like missing a turn in a maze – you might end up in the wrong place altogether! So, pay close attention to those minus signs in front of parentheses, and make sure you distribute them carefully. Once we've done this, the expression will be ready for the next step, which is where the real fun begins: combining like terms and simplifying the whole thing down to its most basic form.
Applying this to our expression, (1.6x + 7.3) + (-0.6x + 2) - (7.8 - 3.4x) becomes 1.6x + 7.3 - 0.6x + 2 - 7.8 + 3.4x. Notice how the terms inside the last set of parentheses had their signs changed? The 7.8 became -7.8, and the -3.4x became +3.4x. This is the power of distributing that negative sign! Now that we've gotten rid of those pesky parentheses, we're ready to move on to the next stage of our mission: gathering our like terms and combining them into a single, simplified expression. This is where the expression really starts to take shape, and we can see the result of all our hard work. So, let's keep going – we're almost there!
Combining Like Terms
Now comes the fun part – combining like terms! Like terms are those that have the same variable raised to the same power (in this case, 'x' to the power of 1) or are constants (just numbers). So, in our expression 1.6x + 7.3 - 0.6x + 2 - 7.8 + 3.4x, we have 'x' terms (1.6x, -0.6x, and 3.4x) and constant terms (7.3, 2, and -7.8). To combine them, we simply add or subtract their coefficients (the numbers in front of the 'x') and add or subtract the constants. Think of it like sorting your socks – you put all the same kind together! First, let's deal with the 'x' terms. We've got 1.6x, then we subtract 0.6x, and then we add 3.4x. So, 1.6 - 0.6 + 3.4 equals 4.4. That means our combined 'x' term is 4.4x. Next up are the constants. We've got 7.3, then we add 2, and then we subtract 7.8. So, 7.3 + 2 - 7.8 equals 1.5. That means our combined constant term is 1.5. And just like that, we've combined all our like terms! By carefully grouping and adding/subtracting the coefficients and constants, we've simplified the expression significantly. This is a key step in algebra, and mastering it opens the door to solving more complex equations and problems.
The Simplified Expression
After all that work, we've arrived at our simplified expression! We combined the 'x' terms and the constants, and now we can put it all together. From the previous step, we found that the combined 'x' term is 4.4x, and the combined constant term is 1.5. So, we simply add these together to get our final answer: 4.4x + 1.5. And there you have it! We've taken a somewhat complicated expression and simplified it down to its most basic form. This is the power of algebraic manipulation – by following the rules and steps, we can make even the most daunting expressions manageable. This simplified form is much easier to work with if we needed to, say, solve an equation or graph a function. It's like taking a messy room and organizing it – everything is clearer and easier to find. So, congratulations on making it to the end! You've successfully navigated the process of simplifying an algebraic expression, and you've gained a valuable skill that will help you in your mathematical journey. Remember, practice makes perfect, so keep working at it, and you'll become a simplifying pro in no time!
So, the final simplified form of (1.6x + 7.3) + (-0.6x + 2) - (7.8 - 3.4x) is 4.4x + 1.5. Great job, everyone! We took a complex expression and broke it down into manageable steps. Remember, simplifying expressions is a fundamental skill in algebra, and mastering it will set you up for success in more advanced topics. Keep practicing, and you'll become a pro at this in no time!
