Simplifying $-9.2(8x-4) + 0.7(2+6.3x)$: A Step-by-Step Guide

Hey guys! Today, we're diving into a fun little math problem: simplifying the expression $-9.2(8x-4) + 0.7(2+6.3x)$. It might look a bit intimidating at first, but don't worry, we'll break it down step by step so it's super easy to understand. Think of it like untangling a knot – we just need to follow the right steps! This is a common type of problem you'll see in algebra, and mastering it will definitely help you out. We will explore how to correctly apply the distributive property, combine like terms, and arrive at the simplified expression. So, grab your pencils and let's get started!

Understanding the Problem

Before we jump into the solution, let's make sure we understand what the question is asking. We have an algebraic expression with parentheses, variables, and constants. Our goal is to simplify it, meaning we want to rewrite it in a cleaner, more compact form. This usually involves getting rid of the parentheses and combining any terms that are similar. The key here is to follow the order of operations (PEMDAS/BODMAS) and be careful with our arithmetic. Simplifying expressions like this is a fundamental skill in algebra, and it's used in many other areas of mathematics and even in real-world applications. So, let’s dig in and see how it's done!

The Distributive Property: Our First Tool

The first thing we need to tackle is those parentheses. To do that, we'll use the distributive property. Remember, this property tells us that to multiply a single term by an expression inside parentheses, we need to multiply the term by each part of the expression individually. In other words, $a(b + c) = ab + ac$. It's like sharing – the term outside the parentheses gets "shared" with everything inside. Let’s apply this to our expression. We have two sets of parentheses: $-9.2(8x - 4)$ and $0.7(2 + 6.3x)$. We'll distribute the $-9.2$ across the first set and the $0.7$ across the second set. Make sure to pay close attention to the signs – a negative times a negative is a positive!

Step-by-Step Distribution

Let's break down the distribution step by step. First, we'll distribute the $-9.2$ across $(8x - 4)$: $-9.2 * 8x = -73.6x$ and $-9.2 * -4 = 36.8$. So, $-9.2(8x - 4)$ becomes $-73.6x + 36.8$. Next, we'll distribute the $0.7$ across $(2 + 6.3x)$: $0.7 * 2 = 1.4$ and $0.7 * 6.3x = 4.41x$. So, $0.7(2 + 6.3x)$ becomes $1.4 + 4.41x$. Now, we've successfully eliminated the parentheses! Our expression now looks like this: $-73.6x + 36.8 + 1.4 + 4.41x$. It's starting to look much simpler already, right? The most important thing is to take your time and double-check your multiplications, especially when dealing with decimals and negative signs.

Combining Like Terms

Okay, we've gotten rid of the parentheses, which is a huge step. Now, we need to combine like terms. What does that mean? Well, like terms are terms that have the same variable raised to the same power. In our expression, $-73.6x$ and $4.41x$ are like terms because they both have the variable $x$ to the power of 1. Similarly, $36.8$ and $1.4$ are like terms because they are both constants (numbers without any variables). Combining like terms is like grouping similar objects together to count them more easily. We simply add or subtract the coefficients (the numbers in front of the variables) of the like terms. So, let’s gather our like terms and see what we get.

Grouping and Simplifying

Let's rewrite our expression, grouping the like terms together: $(-73.6x + 4.41x) + (36.8 + 1.4)$. Now, we can perform the addition and subtraction within each group. For the $x$ terms, we have $-73.6x + 4.41x$. To combine these, we subtract $4.41$ from $73.6$ and keep the negative sign since $-73.6$ has a larger absolute value. This gives us $-69.19x$. For the constants, we simply add $36.8$ and $1.4$, which gives us $38.2$. So, our expression now looks like this: $-69.19x + 38.2$. And guess what? We've simplified it as much as we can! There are no more like terms to combine.

The Final Simplified Expression

Drumroll, please! After all that careful distribution and combining, we've arrived at our final simplified expression: $-69.19x + 38.2$. This is the most compact and clean way to write the original expression. See, it wasn't so scary after all! We successfully navigated through the parentheses, the decimals, and the negative signs. It’s a great feeling when you can take a complex-looking expression and simplify it down to its core components.

Checking Our Work

It's always a good idea to double-check our work, just to make sure we haven't made any silly mistakes along the way. One way to do this is to substitute a value for $x$ into both the original expression and our simplified expression and see if we get the same result. For example, let's try $x = 0$. In the original expression, $-9.2(8(0)-4) + 0.7(2+6.3(0))$, this simplifies to $-9.2(-4) + 0.7(2) = 36.8 + 1.4 = 38.2$. In our simplified expression, $-69.19(0) + 38.2$, this also simplifies to $38.2$. So, at least for $x = 0$, the expressions are equivalent. While this doesn't guarantee we're correct, it gives us more confidence in our answer. Another way to check is to carefully review each step of our work, paying attention to the distribution and combining like terms.

Conclusion: Mastering Simplification

So there you have it! We've successfully simplified the expression $-9.2(8x-4) + 0.7(2+6.3x)$ to $-69.19x + 38.2$. We used the distributive property to eliminate the parentheses and then combined like terms to arrive at our final answer. Remember, the key to simplifying algebraic expressions is to take it one step at a time, be mindful of the order of operations, and double-check your work. Simplifying expressions is a fundamental skill in algebra, and the more you practice, the better you'll get at it. Keep up the great work, and you'll be a simplification pro in no time! And remember, if you ever get stuck, break the problem down into smaller steps and focus on each step individually. You got this! Now go tackle some more problems and show those expressions who's boss!