Simplifying Expressions: A Step-by-Step Guide

Hey math enthusiasts! Today, we're diving into the world of simplifying expressions. More specifically, we're going to tackle a problem that involves distributing and combining like terms. Our goal is to take a complex-looking expression and turn it into its simplest, most equivalent form. This is super important because simplifying expressions is a fundamental skill in algebra and will pop up all over the place as you work through math problems, whether you're dealing with equations, inequalities, or even more advanced concepts. So, let's break down the problem step-by-step and make sure we understand this process perfectly. We will learn how to simplify expressions like $2(3y + 6) - 3(-4 - y)$ and find the correct equivalent expression from the options given.

Understanding the Problem: The Core Concepts

Before we jump into the calculation, let's quickly review the key concepts at play here. The problem we're dealing with involves two main operations: distribution and combining like terms. Firstly, distribution is when we multiply a term outside the parentheses by each term inside the parentheses. This is where the distributive property comes in handy! It tells us that a(b + c) = ab + ac. Secondly, combining like terms is all about grouping together terms that have the same variable raised to the same power. For example, 3y and 5y are like terms, but 3y and 3y² are not. We can combine like terms by adding or subtracting their coefficients (the numbers in front of the variables). Also, remember that a negative sign in front of parentheses needs to be distributed as well. That means you are multiplying by -1.

Now, let's get down to the actual problem. We have the expression $2(3y + 6) - 3(-4 - y)$. We want to simplify this and find the equivalent expression among the answer choices provided. This seems easy but a common mistake is not distributing the -3 correctly, so pay close attention.

Step-by-Step Solution: Unveiling the Simplified Form

Alright, buckle up, guys! We're going to simplify the expression $2(3y + 6) - 3(-4 - y)$ step by step. Here's how to do it:

1. Distribute the 2: We start by multiplying the 2 outside the first set of parentheses by each term inside: $2 * 3y = 6y$ and $2 * 6 = 12$. So, the first part of our expression becomes $6y + 12$.
2. Distribute the -3: Now, we move on to the second set of parentheses. Remember, we're distributing a -3, so be careful with the signs! $-3 * -4 = 12$ and $-3 * -y = +3y$. This gives us $+12 + 3y$.
3. Rewrite the Expression: After distributing, our expression now looks like this: $6y + 12 + 12 + 3y$.
4. Combine Like Terms: Finally, we combine the like terms. We have two 'y' terms: $6y$ and $3y$. Combining them gives us $6y + 3y = 9y$. We also have two constant terms: $12$ and $12$. Combining them gives us $12 + 12 = 24$.
5. Simplified Expression: Putting it all together, the simplified expression is $9y + 24$.

So, the simplified expression of $2(3y + 6) - 3(-4 - y)$ is $9y + 24$.

Matching the Answer: Finding the Correct Choice

Now that we've simplified the expression, let's match our answer to the choices provided. The simplified expression we found is $9y + 24$. Looking at the answer options:

• (A) $9y$
• (B) $5y + 24$
• (C) $9y - 24$
• (D) $9y + 24$

It's clear that option (D) $9y + 24$ is the correct answer. This is the expression we derived through our step-by-step simplification process. So, the final answer is (D).

Why This Matters: The Importance of Simplifying

Why spend so much time simplifying expressions? Well, simplifying is a foundational skill in algebra and beyond. Here are a few reasons why it's super important:

• Solving Equations: Simplifying expressions is often the first step in solving equations. By simplifying each side of an equation, you make it easier to isolate the variable and find the solution. Imagine trying to solve a complicated equation without simplifying first! It would be a nightmare.
• Understanding Relationships: Simplified expressions help you see the relationships between variables more clearly. They make it easier to understand how changes in one variable affect others.
• Reducing Errors: Simplifying reduces the chance of making mistakes. When you work with a simpler expression, there are fewer terms to keep track of, and the chances of a calculation error go down.
• Efficiency: Simplifying makes complex problems more manageable. It's much easier to work with a simplified expression than with a complex one. This saves time and reduces the cognitive load.
• Building Blocks for More Advanced Math: The skill of simplifying expressions is a building block for more complex math concepts, such as factoring, working with polynomials, and calculus. If you can't simplify, you'll struggle with these more advanced topics.

In essence, simplifying expressions is like tidying up your desk before starting a project. It clears the clutter and makes it easier to focus on the task at hand.

Common Mistakes and How to Avoid Them

Let's go over some common mistakes people make when simplifying expressions and how to avoid them:

• Incorrect Distribution: The most common mistake is forgetting to distribute to every term inside the parentheses or making a sign error. Always make sure to multiply the term outside the parentheses by each term inside. Double-check your signs – a negative sign can easily cause problems.
• Combining Unlike Terms: Remember, you can only combine like terms. Don't try to add or subtract terms with different variables or exponents. For instance, you can't combine 3x and 3x², or 2y and 7z. They're not