Solving For 'u': A Step-by-Step Guide

Hey everyone! Today, we're diving into a classic algebra problem: solving for a variable. Specifically, we're going to break down the equation $10 = 2u + 3(u + 5)$ and find out what 'u' equals. Don't worry if algebra isn't your favorite thing – we'll go through it step by step, making it super clear and easy to follow. Get ready to flex those math muscles, because we're about to make this equation our... well, you get the idea! This is going to be a fun journey, and by the end, you'll be able to confidently solve this type of equation on your own. Let's get started, shall we?

Understanding the Basics: What We're Dealing With

Alright, before we jump into the nitty-gritty of solving the equation, let's make sure we're all on the same page. What exactly are we looking at here? Well, we have a simple algebraic equation. An equation is basically a mathematical statement that shows two things are equal. Think of it like a balanced scale; whatever's on one side must equal what's on the other. In our case, the scale is balanced because the expression on the left-hand side (LHS), which is just '10', is equal to the expression on the right-hand side (RHS), which is '2u + 3(u + 5)'.

Our mission? To find the value of 'u' that makes this equation true. 'u' is our unknown, the variable we're trying to isolate. The goal is to manipulate the equation, using the rules of algebra, until we have 'u' all by itself on one side of the equation, with a number on the other side. That number is the solution – the value of 'u' that satisfies the original equation. We'll be using some fundamental algebraic principles such as the distributive property, combining like terms, and inverse operations (addition/subtraction and multiplication/division) to get there. It's like a puzzle, and we're going to solve it piece by piece! The objective here is to give you a strong foundation, so the concepts you learn can be applied to many other problems. Trust me; once you grasp these basics, you'll be well-equipped to tackle more complex equations down the road. So, ready to unlock the secrets of this equation?

This isn't just about memorizing steps; it's about understanding why each step works. This deeper understanding will make solving future problems much easier. Remember, every step we take is designed to simplify the equation and get us closer to our goal: isolating 'u'.

Step-by-Step Solution: Unraveling the Equation

Okay, guys, let's roll up our sleeves and get to work! We're going to break down the equation $10 = 2u + 3(u + 5)$ step by step, making sure every move is crystal clear. Remember, the ultimate goal is to get 'u' by itself on one side of the equation. So, let's start with the first step which is the distributive property. This property allows us to get rid of those pesky parentheses. It states that a(b + c) = ab + ac. In our equation, we have 3(u + 5). We'll multiply the 3 by both terms inside the parentheses. So, 3 times 'u' is 3u, and 3 times 5 is 15. This transforms our equation into $10 = 2u + 3u + 15$.

Now, for the second step, we'll combine like terms. Like terms are terms that have the same variable raised to the same power. In our equation, we have two 'u' terms: 2u and 3u. Combining them means adding their coefficients (the numbers in front of the variable). So, 2u + 3u equals 5u. Our equation now becomes $10 = 5u + 15$.

Next up, we'll isolate the term with 'u'. To do this, we need to get rid of that +15 on the right side of the equation. We do this by performing the inverse operation: subtraction. We'll subtract 15 from both sides of the equation. Remember, whatever you do to one side, you must do to the other to keep the equation balanced. So, we have $10 - 15 = 5u + 15 - 15$. This simplifies to $-5 = 5u$.

Finally, we'll solve for 'u'. We now have $-5 = 5u$. To isolate 'u', we need to get rid of the 5 that's multiplying it. We do this by performing the inverse operation: division. We'll divide both sides of the equation by 5. So, $-5 / 5 = 5u / 5$. This gives us $-1 = u$, or $u = -1$. Congratulations, we've found our solution! We've successfully solved for 'u' in the given equation.

Each step is a building block, gradually simplifying the equation until we isolate the variable. Notice how we use inverse operations to undo the operations around 'u', working our way towards the solution.

Verification: Checking Our Answer

Alright, we've solved for 'u', and we got a value of -1. But how do we know if we're right? Always, always check your work! The best way to make sure our solution is correct is to substitute the value of 'u' back into the original equation and see if it holds true. This is an important step. It's like the final quality check, ensuring that our answer makes sense in the context of the problem.

So, our original equation was $10 = 2u + 3(u + 5)$. We're going to replace every 'u' in this equation with -1 and see if both sides are equal. Let's do it! The equation becomes:

$10 = 2(-1) + 3((-1) + 5)$

Now, let's simplify step by step:

$10 = -2 + 3(4)$

$10 = -2 + 12$

$10 = 10$

And there we have it! Both sides of the equation are equal, which means our solution, u = -1, is correct. Our answer fits perfectly, which means we have done our job properly. It's always a good practice to do this, especially when you are learning. This small step can really help you understand the whole process.

This simple substitution and verification help cement your understanding. Doing this helps confirm the correctness of our solution. This ensures our calculations are correct and boosts our confidence in the result. By verifying our answer, we ensure that we haven't made any mistakes. This is a crucial step in ensuring the integrity of our solution. So, in summary, you should always verify the solution.

Key Takeaways and Tips

Fantastic job, everyone! We've made it to the end, and hopefully, you now feel more confident about solving algebraic equations. Let's recap some key takeaways and tips to help you master this skill. First, understand the order of operations. Remember to follow the order of operations (PEMDAS/BODMAS) when simplifying expressions: Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). This ensures you perform calculations in the correct order, avoiding common mistakes.

Second, practice, practice, practice! The more you solve equations, the more comfortable and proficient you'll become. Start with simple equations and gradually increase the difficulty. Try to find a lot of practice problems. Working through different types of problems helps you recognize patterns and apply the appropriate strategies for solving them.

Third, always check your work. Substituting your solution back into the original equation is crucial to verify its accuracy. This simple step can save you from making errors and reinforce your understanding. Always take that extra minute or two to ensure the answer works. Make sure to double-check everything, it helps make sure everything is perfect.

Fourth, break down complex problems. Complex problems seem intimidating. Break them down into smaller, manageable steps. This makes the process less overwhelming and allows you to focus on one aspect at a time. This method applies to many aspects of life. Breaking down complex tasks into smaller, more manageable steps makes them less daunting and helps maintain focus.

Fifth, don't be afraid to ask for help. If you're stuck, don't hesitate to ask your teacher, classmates, or online resources for assistance. Seeking help is a sign of intelligence, not weakness. There are tons of resources available.

Finally, believe in yourself! With consistent effort and a positive attitude, you can definitely master algebra and other math concepts. It's all about persistence and the willingness to learn. Keep practicing and keep up the great work!

I really hope this guide helped you guys! Solving for 'u' is a fundamental skill. The more you practice, the easier it becomes. You've got this! Keep practicing, and you will see amazing results. I have absolute faith that you can do this.