Simplify 5(y-1)+2(4y+3): Easy Math Guide

Hey everyone! Today, we're diving deep into the awesome world of algebra to tackle a common problem: simplifying and expanding expressions. Specifically, we're going to break down how to simplify the expression $5(y-1)+2(4 y+3)$. Don't worry if algebra seems a bit daunting; we'll go through it step-by-step, making it super easy to understand. By the end of this, you'll be a pro at simplifying these kinds of expressions, and you'll see just how cool and logical math can be. So grab your notebooks, get comfy, and let's get this math party started!

Understanding the Basics: What Does "Simplify and Expand" Mean?

Alright guys, before we jump into our specific problem, let's quickly chat about what it means to "simplify and expand" an algebraic expression. Think of it like tidying up a messy room. You have different toys (terms) scattered around, and some might be grouped together in boxes (parentheses). Our job is to put everything in its right place, combine similar items, and make the whole room look neat and organized.

Expanding basically means getting rid of those pesky parentheses. We do this by using the distributive property. Remember that? It's like when you have a number outside a set of parentheses, and you multiply that number by everything inside the parentheses. So, if you see something like $a(b+c)$, it becomes $ab + ac$. Pretty neat, right? We're essentially spreading that outside number to each term inside.

Simplifying is the next step after expanding. Once we've distributed everything, we might have several terms that are alike. For example, if we end up with a bunch of 'y' terms and a bunch of number terms, simplifying means we combine all the 'y' terms together and all the number terms together. This makes our expression much shorter and easier to work with. So, if we had $3y + 5y + 2 + 7$, we'd combine the 'y's to get $8y$ and the numbers to get $9$, resulting in $8y + 9$. See? Much cleaner!

Our goal with the expression $5(y-1)+2(4 y+3)$ is to perform both these actions. We'll first expand by distributing the numbers outside the parentheses, and then we'll simplify by combining like terms. It’s a two-step process that’s fundamental to algebra. Mastering this will unlock a lot of other cool math concepts down the line. So, let's get ready to apply these principles to our specific problem and make it sparkle!

Step 1: Expanding the Expression Using the Distributive Property

Okay, team, let's tackle the first part: expanding our expression, $5(y-1)+2(4 y+3)$. This is where the distributive property comes into play, and it's a crucial skill in algebra. Remember, the distributive property says that when you have a number multiplied by a sum or difference inside parentheses, you multiply that number by each term inside the parentheses. Let's break it down piece by piece.

First, look at the term $5(y-1)$. The number $5$ is outside the parentheses $(y-1)$. So, we need to multiply $5$ by both $y$ and $-1$.

• $5 imes y = 5y$
• $5 imes (-1) = -5$

So, $5(y-1)$ expands to $5y - 5$. Easy peasy!

Now, let's move to the second part of our expression: $2(4 y+3)$. The number $2$ is outside the parentheses $(4y+3)$. We'll distribute the $2$ to both $4y$ and $3$.

• $2 imes (4y) = 8y$
• $2 imes 3 = 6$

So, $2(4y+3)$ expands to $8y + 6$.

Now, we need to put these expanded parts back together with the addition sign that was between them in the original expression. Our original expression was $5(y-1)+2(4 y+3)$. After expanding each part, it becomes:

(5y - 5) + (8y + 6)

At this stage, we've successfully expanded the expression. We've gotten rid of the parentheses, which is a big step! You've applied the distributive property correctly, which is a key algebraic maneuver. Now, the expression looks a bit less intimidating, and we're ready for the next phase: simplifying by combining like terms. Keep up the great work, guys!

Step 2: Simplifying by Combining Like Terms

Alright, awesome mathematicians, we've conquered the expansion phase! We now have the expression $(5y - 5) + (8y + 6)$. The next crucial step is to simplify it by combining what we call "like terms." Think of "like terms" as things that are the same kind – in algebra, this usually means terms that have the same variable raised to the same power, or terms that are just plain numbers (constants).

In our expression, we have two types of terms: terms with the variable '$y$' and terms that are just numbers.

Let's first identify all the terms that have '$y$' in them. We have $5y$ from the first part and $8y$ from the second part. These are our 'y' terms. To combine them, we simply add their coefficients (the numbers in front of the 'y').

• $5y + 8y = (5 + 8)y = **13y**$

Great! We've combined all the 'y' terms into a single term, $13y$. This is a major simplification.

Now, let's look at the terms that are just numbers (constants). We have $-5$ from the first part and $+6$ from the second part. To combine these, we add them together.

• $-5 + 6 = **1**$

Fantastic! We've combined the constant terms into a single number, $1$.

Finally, we put our combined 'y' term and our combined constant term back together to form our final simplified expression. Remember, we had $13y$ and $1$. Since one is a 'y' term and the other is a constant, they are not like terms, so we can't combine them further. We just write them next to each other.

The simplified expression is $13y + 1$.

You guys did it! We took an expression that looked a bit complex, expanded it using the distributive property, and then simplified it by combining like terms. The result, $13y + 1$, is the most simplified form of the original expression $5(y-1)+2(4 y+3)$. This process is super useful in solving equations and understanding more advanced algebra.

Putting It All Together: The Final Simplified Expression

So, let's recap what we've done, guys. We started with the expression $5(y-1)+2(4 y+3)$. Our mission was to expand and simplify it.

First, we used the distributive property to expand. We multiplied $5$ by $(y-1)$ to get $5y - 5$. Then, we multiplied $2$ by $(4y+3)$ to get $8y + 6$. This gave us a new expression: $(5y - 5) + (8y + 6)$.

Next, we combined like terms. We grouped the '$y$' terms together: $5y + 8y$, which equals $13y$. We also grouped the constant terms together: $-5 + 6$, which equals $1$.

Finally, we combined these results to get our final simplified expression: $13y + 1$.

This is the answer! It's the most concise way to represent the original expression. Every step we took was logical and followed the rules of algebra. Remember, expanding helps to remove parentheses, and simplifying by combining like terms makes the expression as short and manageable as possible. These are fundamental techniques that you'll use constantly in your math journey. Keep practicing these steps, and you'll become a simplification ninja in no time! You've got this!

Why is Simplifying Expressions Important?

Now, you might be thinking, "Okay, that was cool, but why do we even bother simplifying expressions like $5(y-1)+2(4 y+3)$?" That's a totally valid question, guys! The ability to simplify algebraic expressions is a cornerstone of mathematics, and it's incredibly useful for several reasons.

Firstly, clarity and readability. Imagine trying to solve a complex problem with a long, messy expression versus a short, neat one. Simplifying makes expressions much easier to understand and work with. It's like decluttering your workspace – when everything is organized, you can focus on the task at hand without getting bogged down in unnecessary details. The simplified form, $13y + 1$, is much easier to grasp than the original $5(y-1)+2(4 y+3)$.

Secondly, solving equations. Many math problems involve solving equations, where you need to find the value of an unknown variable (like '$y$' in our case). Before you can isolate and solve for the variable, you almost always need to simplify both sides of the equation. If you have an equation like $5(y-1)+2(4 y+3) = 27$, you can't easily solve for '$y$' until you simplify the left side to $13y + 1$. Then the equation becomes $13y + 1 = 27$, which is much simpler to solve.

Thirdly, reducing errors. When expressions are long and complex, there's a higher chance of making mistakes during calculations. Simplifying an expression before you start plugging in values or performing further operations significantly reduces the risk of errors. By combining like terms and distributing correctly, we ensure accuracy.

Finally, foundational skill for higher math. Whether you're heading into calculus, physics, engineering, or computer science, you'll be dealing with algebraic expressions constantly. Understanding how to manipulate and simplify them is a prerequisite for tackling more advanced concepts. It's like learning your ABCs before you can read a novel.

So, the next time you're asked to simplify an expression, remember that you're not just doing busywork. You're building essential skills that will serve you well in all your future mathematical and scientific endeavors. Keep practicing, and you'll master it!

Practice Makes Perfect: Another Example!

To really nail this down, let's try another example together. This will reinforce the steps we've learned and boost your confidence. Suppose we need to simplify the expression $3(2x + 5) - 4(x - 2)$. Ready? Let's dive in!

Step 1: Expand using the distributive property.

• For the first part, $3(2x + 5)$: $3 imes 2x = 6x$ and $3 imes 5 = 15$. So, this part becomes $6x + 15$.
• For the second part, $-4(x - 2)$: Be super careful with the negative sign here! $-4 imes x = -4x$ and $-4 imes (-2) = +8$. So, this part becomes $-4x + 8$.

Now, let's put them together: $(6x + 15) + (-4x + 8)$. Notice how we kept the negative sign with the $4x$. This is crucial!

Step 2: Combine like terms.

• Combine the '$x$' terms: $6x + (-4x) = 6x - 4x = **2x**$.
• Combine the constant terms: $15 + 8 = **23**$.

Step 3: Write the final simplified expression.

Putting our combined terms together, we get $2x + 23$.

See? You just simplified another expression! The key is to be methodical: distribute carefully (especially with negative signs) and then group and combine your like terms. The more you practice, the faster and more accurate you'll become. Keep up the awesome work!

Conclusion: You've Mastered Expression Simplification!

Awesome job, everyone! We've successfully taken the expression $5(y-1)+2(4 y+3)$, expanded it using the distributive property, and then simplified it by combining like terms. The final, neat, and tidy answer is $13y + 1$. You've walked through the process step-by-step, from understanding what expanding and simplifying mean to applying these concepts to solve the problem and even looking at why it's so important.

Remember the key steps:

1. Distribute: Multiply the number outside the parentheses by each term inside.
2. Combine Like Terms: Group and add or subtract terms with the same variable and the same exponent, and do the same for the constant terms.

These skills are fundamental building blocks in algebra. They might seem simple now, but they unlock the door to solving much more complex problems, understanding advanced math, and succeeding in various scientific fields. Never underestimate the power of mastering these basics!

Keep practicing, tackle new problems, and don't be afraid to ask questions. Every expression you simplify makes you a stronger mathematician. You've got this, and I'm excited for your next math adventure!