Solve For W: 5(2w+4)=4(2w+9)

Hey guys, welcome back to Plastik Magazine! Today, we're diving deep into the fascinating world of mathematics to tackle a specific algebraic equation. We're going to unravel the mystery behind "What is the solution to the equation $5(2 w+4)=4(2 w+9)$?" Don't worry if algebra gives you the jitters; we're going to break it down step-by-step, making it super easy to understand. Our goal is to find the value of 'w' that makes this equation true. This is a fundamental skill in math, and once you get the hang of it, you'll feel like a total math whiz! We'll explore the properties of equality and how to isolate the variable 'w' using inverse operations. By the end of this article, you'll not only know the answer but also understand how we got there, empowering you to solve similar equations with confidence. So, grab your thinking caps, and let's get started on this mathematical adventure!

Understanding the Equation: A Closer Look at $5(2w+4)=4(2w+9)$

Alright, let's start by really looking at the equation we've got: $5(2 w+4)=4(2 w+9)$. The first thing you'll notice is that we have parentheses on both sides of the equal sign. This means we need to use the distributive property to simplify things. The distributive property is like a magic spell that lets us multiply a number outside the parentheses by each term inside. So, on the left side, we're going to multiply 5 by both $2w$ and 4. On the right side, we'll multiply 4 by both $2w$ and 9. This process is crucial because it removes the parentheses and sets us up to solve for 'w'. It's important to be super careful with your multiplication here – a small slip-up can lead to a completely different answer! Remember, the distributive property states that $a(b+c) = ab + ac$. Applying this to our equation, the left side becomes $5 \times 2w + 5 \times 4$, which simplifies to $10w + 20$. Similarly, the right side becomes $4 \times 2w + 4 \times 9$, which simplifies to $8w + 36$. So, our equation is now transformed into a much more manageable form: $10w + 20 = 8w + 36$. This is a significant step because we've eliminated the parentheses and have a linear equation with terms involving 'w' and constant terms on both sides. Getting to this stage is all about applying basic algebraic rules correctly, and it's a testament to the power of systematic simplification in mathematics. We're essentially rewriting the original equation in an equivalent, but simpler, form, which is key to solving for our unknown variable.

Isolating the Variable: Getting 'w' by Itself

Now that we have our simplified equation, $10w + 20 = 8w + 36$, our next mission is to get all the terms with 'w' on one side of the equation and all the constant terms on the other side. Think of it like sorting your socks – you want all the blue ones together and all the red ones together. To do this, we use inverse operations. Remember, whatever you do to one side of the equation, you must do to the other side to keep it balanced. Our goal is to isolate 'w'. Let's start by moving the 'w' terms. We have $10w$ on the left and $8w$ on the right. To get rid of the $8w$ on the right, we'll subtract $8w$ from both sides. So, we have $10w + 20 - 8w = 8w + 36 - 8w$. This simplifies to $2w + 20 = 36$. See? We've successfully moved all the 'w' terms to the left side! Now, we need to move the constant terms. We have $+20$ on the left side with our 'w' term. To get rid of it, we'll subtract 20 from both sides of the equation: $2w + 20 - 20 = 36 - 20$. This leaves us with $2w = 16$. We're so close, guys! We've managed to group all the 'w' terms on one side and all the numbers on the other. This process of moving terms across the equals sign by performing the opposite operation is the cornerstone of solving linear equations. It's all about maintaining equality while systematically simplifying the expression to reveal the value of the unknown. Each step builds upon the last, guiding us closer to the final answer.

Finding the Solution: The Final Calculation

We've reached the final stretch, folks! Our equation is now $2w = 16$. We're just one step away from finding the value of 'w'. Currently, 'w' is being multiplied by 2. To isolate 'w', we need to perform the inverse operation of multiplication, which is division. We will divide both sides of the equation by 2: $\frac{2w}{2} = \frac{16}{2}$. Performing this division, we get $w = 8$. And there you have it! The solution to the equation $5(2 w+4)=4(2 w+9)$ is $w=8$. This final step is where all our previous hard work pays off. By carefully applying the distributive property and using inverse operations to isolate the variable, we've successfully determined the value of 'w' that satisfies the original equation. It's a systematic process that, when followed correctly, always leads to the right answer. So, the answer is (B) $w=8$. Remember, the key takeaways here are the distributive property and the concept of balancing the equation using inverse operations. Keep practicing these skills, and you'll be solving equations like a pro in no time!

Checking Your Work: Ensuring Accuracy

It's always a good idea, especially in mathematics, to check your work to make sure you haven't made any silly mistakes. This step is super important and will give you confidence in your answer. We found that the solution to the equation $5(2 w+4)=4(2 w+9)$ is $w=8$. To check this, we substitute $w=8$ back into the original equation and see if both sides are equal. Let's plug it in:

On the left side: $5(2(8) + 4) = 5(16 + 4) = 5(20) = 100$.

On the right side: $4(2(8) + 9) = 4(16 + 9) = 4(25) = 100$.

Since the left side (100) equals the right side (100), our solution $w=8$ is correct! This verification process is a powerful tool in algebra. It confirms that the value we found for 'w' truly makes the original statement of equality valid. It’s like a double-check to ensure that all the steps we took, from distributing to isolating the variable, were performed accurately. This meticulous approach to problem-solving is what distinguishes a good mathematician from a great one. It instills a sense of thoroughness and accuracy, qualities that are invaluable not just in math but in all aspects of life. So, never skip the check step – it’s your best friend in confirming your solutions and building your mathematical prowess. It reinforces the understanding that equations represent a delicate balance, and any solution must respect that balance perfectly.

Conclusion: Mastering Algebraic Equations

So there you have it, team! We've successfully tackled the equation $5(2 w+4)=4(2 w+9)$ and found that the solution is $w=8$, which corresponds to option (B). We walked through the entire process, from using the distributive property to simplify the equation, to isolating the variable 'w' using inverse operations, and finally, checking our answer to ensure accuracy. Understanding how to solve linear equations like this one is a fundamental building block in mathematics. It's a skill that opens the door to more complex algebraic problems and concepts. Remember the key steps: 1. Distribute: Clear the parentheses by multiplying the term outside by each term inside. 2. Combine Like Terms: Move all terms containing the variable to one side and all constant terms to the other side using inverse operations. 3. Isolate the Variable: Perform the final inverse operation to find the value of the variable. 4. Check Your Work: Substitute your solution back into the original equation to verify. Mastering these techniques will not only help you ace your math tests but also equip you with critical thinking and problem-solving skills that are transferable to countless real-world situations. Keep practicing, stay curious, and don't be afraid to tackle challenging problems. The more you practice, the more comfortable and confident you'll become with algebra. Happy solving!