Simplify $5(2y + 2) - 14$ A Step-by-Step Guide

Understanding the Expression

Hey guys! Let's dive into simplifying this algebraic expression: $5(2y + 2) - 14$. At first glance, it might seem a bit intimidating, but don't worry! We're going to break it down step by step, making it super easy to understand. Our goal here is to take this expression and rewrite it in its simplest form, meaning we want to combine any like terms and get rid of those parentheses. Think of it like decluttering your room – we're just tidying up the math! Simplifying expressions is a fundamental skill in algebra, and mastering it will help you tackle more complex problems down the road. We'll be using the distributive property and combining like terms, so if those concepts sound a little fuzzy, no sweat – we'll go through them together. Remember, math is like building with LEGOs; each piece (or step) builds on the last. By the end of this, you'll not only be able to simplify this particular expression but also have a solid understanding of how to approach similar problems. So, grab your mental toolkit, and let's get started!

The first thing we notice is the $5$ outside the parentheses, and that's our cue to use the distributive property. What this means is that we need to multiply the $5$ by each term inside the parentheses. So, we're going to multiply $5$ by $2y$ and then $5$ by $2$. This is like sharing the $5$ with everyone inside the parentheses. When we multiply $5$ by $2y$, we get $10y$. Think of it as having five groups of $2y$, which gives us a total of $10y$. Next, we multiply $5$ by $2$, which is a straightforward $10$. So, after distributing, our expression looks like this: $10y + 10 - 14$. Notice how the parentheses are gone? That's the magic of the distributive property! We've successfully expanded the expression, and now we're one step closer to simplifying it. Remember, the distributive property is a key tool in algebra, and it's all about making sure everything outside the parentheses gets properly multiplied with everything inside. Now that we've handled the distribution, let's move on to the next step: combining like terms.

Applying the Distributive Property

The crucial first step in simplifying the expression $5(2y + 2) - 14$ involves applying the distributive property. This property is a fundamental concept in algebra, allowing us to eliminate parentheses and make the expression easier to manage. Basically, the distributive property states that $a(b + c) = ab + ac$. In our case, the $5$ outside the parentheses needs to be multiplied by both terms inside: $2y$ and $2$. This means we're going to perform two multiplications: $5 * 2y$ and $5 * 2$. Let's tackle the first one: $5 * 2y$. When multiplying a constant by a term with a variable, we multiply the constants together. So, $5 * 2$ equals $10$, and we keep the $y$. This gives us $10y$. Now, let's move on to the second multiplication: $5 * 2$. This one is a simple arithmetic calculation, and $5 * 2$ equals $10$. So, after applying the distributive property, the expression $5(2y + 2)$ becomes $10y + 10$. We've successfully distributed the $5$, and the parentheses are gone! But remember, we still have the $-14$ at the end of the original expression. So, let's bring that down. Our expression now looks like this: $10y + 10 - 14$. We've made significant progress, but we're not quite finished yet. The next step is to combine the like terms, which will further simplify our expression. Understanding and applying the distributive property is essential for simplifying algebraic expressions, and you've just nailed it!

Combining Like Terms

Okay, guys, we've made it to the next exciting part: combining like terms! After applying the distributive property, our expression looks like this: $10y + 10 - 14$. Now, what exactly are “like terms”? Well, like terms are terms that have the same variable raised to the same power. In our expression, $10y$ is a term with the variable $y$, and $10$ and $-14$ are constant terms (they don't have any variables). The constants are like terms because they're just numbers, and we can combine them. So, we're going to focus on combining the $10$ and the $-14$. Think of it as adding and subtracting regular numbers. We have $10 - 14$, which equals $-4$. It's like you have $10$ dollars and you owe $14$ dollars; you end up being $4$ dollars in debt. Now, let's put it all together. We have $10y$ (which doesn't have any like terms to combine with), and we've combined $10$ and $-14$ to get $-4$. So, our simplified expression is $10y - 4$. And that's it! We've successfully simplified the expression by combining the like terms. Notice how much cleaner and simpler it looks now? We started with $5(2y + 2) - 14$, and after distributing and combining like terms, we arrived at $10y - 4$. This is the simplified form of the expression. Remember, combining like terms is all about grouping together the terms that are similar, making the expression as concise as possible. You've done a fantastic job! We're one step closer to mastering algebraic simplification.

The Final Simplified Form

Alright, we've reached the finish line! After all our hard work, distributing and combining like terms, we've arrived at the final simplified form of the expression: $10y - 4$. Isn't it satisfying to see how a complex-looking expression can be boiled down to something so clean and simple? Let's quickly recap what we did. We started with $5(2y + 2) - 14$. First, we applied the distributive property, multiplying the $5$ by both terms inside the parentheses: $2y$ and $2$. This gave us $10y + 10$. Then, we brought down the $-14$, so our expression became $10y + 10 - 14$. Next, we identified the like terms, which were the constants $10$ and $-14$. We combined these like terms by subtracting $14$ from $10$, resulting in $-4$. Finally, we put it all together, and our simplified expression is $10y - 4$. This expression is now in its simplest form because there are no more like terms to combine, and there are no parentheses. We can't simplify it any further. This final form, $10y - 4$, is much easier to work with and understand than the original expression. You've successfully navigated the process of simplifying an algebraic expression, and you should be proud of yourself! Remember, simplification is a crucial skill in algebra, and you've just added another tool to your math toolbox. Keep practicing, and you'll become a simplification pro in no time!

Practice Problems

Now that we've simplified the expression $5(2y + 2) - 14$, let's put your new skills to the test with some practice problems! Working through additional examples is the best way to solidify your understanding and build confidence. So, grab a pencil and paper, and let's dive in! Remember, the key is to follow the same steps we used in the example: first, apply the distributive property, and then combine like terms. Don't be afraid to make mistakes – that's how we learn! And if you get stuck, just go back and review the steps we took in the example. Let's start with a similar expression: $3(4x - 1) + 7$. Can you simplify this one? What about $2(5a + 3) - 9$? Or maybe a slightly more challenging one: $4(2b - 5) + 3b$? These practice problems are designed to give you a variety of scenarios to work with, helping you master the art of simplification. The more you practice, the more comfortable you'll become with the distributive property and combining like terms. And remember, math isn't about memorizing formulas; it's about understanding the process and applying it in different situations. So, take your time, work through each problem step by step, and celebrate your progress along the way. You've got this! Keep practicing, and you'll become a simplification superstar!