Simplify 4(1 - 2b) + 7b - 10: A Step-by-Step Guide

Hey guys! Today, we're diving into simplifying algebraic expressions. It might seem daunting at first, but trust me, once you break it down, it's totally manageable. We're going to tackle the expression 4(1 - 2b) + 7b - 10 step-by-step, making sure we understand each move we make. So, grab your pencils and paper, and let's get started!

Understanding the Order of Operations

Before we jump into the expression, it's super important to remember our good ol' friend, the order of operations – often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). This is our roadmap for simplifying expressions correctly. Think of it as the golden rule of math expressions!

• Parentheses first: We deal with anything inside parentheses before anything else.
• Exponents next: We handle any exponents or powers.
• Multiplication and Division: These operations come next, and we work them from left to right.
• Addition and Subtraction: Finally, we handle addition and subtraction, also working from left to right.

Why is this order so important, you ask? Well, imagine if we just started adding and subtracting without considering the multiplication. We'd end up with a completely different answer! PEMDAS ensures we all arrive at the same, correct solution. So, keep this order in mind as we tackle the expression 4(1 - 2b) + 7b - 10. It’s the key to unlocking the correct answer and avoiding any mathematical mishaps.

Step 1: Distribute the 4

The first thing we need to do, according to PEMDAS, is to take care of the parentheses. We have 4(1 - 2b). This means we need to distribute the 4 to both terms inside the parentheses. Distribution, in simple terms, means multiplying the term outside the parentheses by each term inside. It's like the 4 is saying, "Hey, I want to multiply with everyone inside!"

So, we multiply 4 by 1, which gives us 4. Then, we multiply 4 by -2b, which gives us -8b. Remember, when we multiply a positive number by a negative number, the result is negative. It's a common mistake to forget the negative sign, so always double-check! This gives us a new, simplified piece of our expression: 4 - 8b. We've successfully distributed the 4 and taken a significant step toward simplifying the entire expression.

Distribution is a fundamental concept in algebra, and it’s used all the time. Mastering this skill is super important for solving more complex equations and expressions down the road. Think of it as building a strong foundation for your algebra knowledge. Each time you practice distribution, you're strengthening that foundation, making it easier to tackle more challenging problems. So, always remember to distribute carefully and pay attention to the signs – it can make a big difference in the final answer!

Step 2: Rewrite the Expression

Now that we've distributed the 4, let's rewrite the entire expression with our simplified piece. Originally, we had 4(1 - 2b) + 7b - 10. After distributing, we know that 4(1 - 2b) is the same as 4 - 8b. So, we can replace the original part with this new, simplified version. Our expression now looks like this: 4 - 8b + 7b - 10. See how we just swapped out the original part with its simplified form? It's like replacing a complicated puzzle piece with a simpler one that fits perfectly.

Rewriting the expression is a crucial step because it helps us visualize the next steps more clearly. It's like decluttering your workspace before starting a new project – you want everything organized and easy to access. By rewriting, we've essentially organized our expression, making it easier to see which terms we can combine. This sets us up perfectly for the next step, which is combining like terms. The goal here is to make the expression as simple and manageable as possible before we move forward. It’s all about being strategic and setting ourselves up for success!

Step 3: Combine Like Terms

Alright, guys, this is where the magic happens! We're going to combine like terms. Now, what exactly are like terms? They're terms that have the same variable raised to the same power. Think of them as buddies that can hang out and combine their values. In our expression, 4 - 8b + 7b - 10, we have two types of terms: constant terms (numbers without variables) and terms with the variable 'b'.

Our constant terms are 4 and -10. We can combine these by simply adding them together: 4 + (-10) = -6. Easy peasy! Next, we have the terms with the variable 'b': -8b and +7b. We combine these by adding their coefficients (the numbers in front of the variable): -8 + 7 = -1. So, -8b + 7b becomes -1b, which we can also write as just -b.

By combining like terms, we've simplified our expression significantly. It's like tidying up a room – we're grouping similar items together to make things neat and organized. This step is essential for simplifying expressions because it reduces the number of terms and makes the expression easier to understand and work with. Remember, like terms are your friends – they make simplifying so much easier!

Step 4: Write the Simplified Expression

Okay, we're in the home stretch now! After combining like terms, we have -6 and -b. To write the simplified expression, we simply put these terms together. Our simplified expression is -b - 6. That's it! We've taken the original expression, 4(1 - 2b) + 7b - 10, and simplified it down to its most basic form. Give yourselves a pat on the back – you've done it!

Writing the simplified expression is the final step in our journey, and it's super satisfying to see how much we've reduced the expression. We started with a longer, more complex expression, and through the magic of distribution and combining like terms, we've arrived at a much simpler version. This is the power of algebra – taking something complicated and breaking it down into its simplest parts. The simplified expression, -b - 6, is much easier to work with if we were to, say, solve an equation or graph a line. So, remember, simplifying isn't just about making things look neater; it's about making the math easier too!

Final Answer

So, to recap, we started with the expression 4(1 - 2b) + 7b - 10. We distributed the 4, rewrote the expression, combined like terms, and finally, arrived at our simplified answer: -b - 6. You guys are rockstars! Simplifying expressions is a fundamental skill in algebra, and you've just nailed it. Keep practicing, and you'll become a simplification pro in no time! Remember, each time you solve a problem like this, you're building your math muscles and becoming more confident in your abilities. So, keep up the great work, and happy simplifying!