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

Hey math enthusiasts! Today, we're diving into a classic algebra problem: solving for a variable. Specifically, we're going to solve for u in the equation 1 = 5(u - 5) + 8u. Don't worry if it looks a little intimidating at first. We'll break it down into easy-to-follow steps. This guide is designed to help anyone from a student just starting out with algebra to someone brushing up on their skills. So, grab your pencils and let's get started. Solving equations is a fundamental skill in mathematics, and understanding how to manipulate and isolate variables is crucial for tackling more complex problems down the line. We'll go through the process systematically, explaining each step to ensure you grasp the underlying principles. Think of this as your personal algebra tutor, guiding you through every twist and turn of the equation. By the end of this, you'll not only have the correct answer but also a solid understanding of the techniques involved. This problem is an excellent example of a linear equation, which means the highest power of the variable u is 1. We'll utilize basic arithmetic operations like distribution, addition, and subtraction to isolate u on one side of the equation. Remember, the goal is always to get u by itself. We're essentially working backward, undoing the operations that are applied to u until it stands alone. So, let's roll up our sleeves and get to work – it's going to be fun!

Step-by-Step Solution

Alright, let's get down to business! Here’s how we'll solve the equation 1 = 5(u - 5) + 8u step by step. We'll make sure that each stage is clear and concise. This method makes it easy to follow and replicate for other algebraic problems. First, we tackle the parentheses. When we're given an equation with parentheses, our first order of business is usually to get rid of them. To do this, we'll use the distributive property. The distributive property says that a(b + c) = ab + ac. In our case, it's 5(u - 5). So, we multiply both u and -5 by 5: 5 * u = 5u and 5 * -5 = -25. This gives us 5u - 25. Now our equation looks like this: 1 = 5u - 25 + 8u.

Now that we've taken care of the parentheses, let's simplify further. Combining like terms is the next step to simplifying our equation. Combine like terms. We look for terms that have the same variable (or no variable at all) and combine them. In our current equation, we have two terms with the variable u: 5u and 8u. Adding them together gives us 13u. So, our equation becomes: 1 = 13u - 25. See how much cleaner that looks? Remember, combining like terms is all about making the equation easier to handle by grouping similar elements together. This simplifies the equation significantly and brings us closer to isolating the variable we want to find out. Now, we are ready to take a closer look at isolating u.

Next, isolate the variable term. Our aim is to get all the terms containing u on one side of the equation and all the constant terms (numbers without variables) on the other side. To do this, we need to get rid of the -25 on the right side. We do this by adding 25 to both sides of the equation. Remember, whatever you do to one side of an equation, you must do to the other side to keep it balanced. So, we add 25 to the left side as well: 1 + 25 = 13u - 25 + 25. This simplifies to 26 = 13u. At this point, the equation has moved from its previous state to another state, that is more suitable for solving for u.

Finally, solve for u. We’re almost there! We have 26 = 13u. To isolate u, we need to get rid of the 13 that's multiplying it. We do this by dividing both sides of the equation by 13. This gives us 26 / 13 = 13u / 13. Which simplifies to 2 = u. Or, u = 2. And there you have it – we've solved for u!

Detailed Breakdown

• Original Equation: 1 = 5(u - 5) + 8u
• Step 1 (Distributive Property): 1 = 5u - 25 + 8u
• Step 2 (Combine Like Terms): 1 = 13u - 25
• Step 3 (Isolate Variable Term): 1 + 25 = 13u
• Step 4 (Simplify): 26 = 13u
• Step 5 (Solve for u): u = 26 / 13
• Final Answer: u = 2

Verification

It's always a good idea to check your work, right? So, let's plug our answer, u = 2, back into the original equation to make sure it's correct. We'll substitute u with 2: 1 = 5(2 - 5) + 8(2). Now let's simplify. 2 - 5 = -3, so we have 1 = 5(-3) + 8(2). Next, 5 * -3 = -15 and 8 * 2 = 16. So now our equation is 1 = -15 + 16. Finally, -15 + 16 = 1. Therefore, 1 = 1. Since the equation holds true, our solution, u = 2, is correct! This is a great way to confirm that all the steps taken to solve the problem have been followed correctly. Verifying your solution not only confirms that you have the right answer but also helps you to understand the equation better.

Tips and Tricks

Staying organized is the name of the game when it comes to solving equations. Keep your work neat and clearly labeled. This will help you avoid mistakes and make it easier to go back and check your work. Write each step on a new line. Use parentheses correctly, especially when dealing with the distributive property. It's easy to miss a term if you're not careful. Double-check your signs – a small mistake with a plus or minus sign can change the entire answer. Make sure you combine like terms properly. If you're struggling, break the problem down into smaller steps. Don't be afraid to take your time and work through each part carefully. Practice makes perfect. The more you solve equations, the more comfortable and confident you'll become. Solve different types of equations to build your skillset.

Common Mistakes and How to Avoid Them

One common mistake is incorrectly applying the distributive property. Remember to multiply every term inside the parentheses by the factor outside. Another common mistake is making errors with signs. Always pay close attention to positive and negative signs. Make sure you combine like terms accurately. Avoid skipping steps. Skipping steps may seem faster, but it increases the risk of making a mistake. Always verify your answer. This is the best way to catch any errors and ensure your solution is correct. By keeping these tips in mind, you will find it easier to work through such problems.

Conclusion

Congratulations, guys! You've successfully solved for u in the equation 1 = 5(u - 5) + 8u. We've gone through the process step-by-step, including a verification to confirm our answer. Remember, solving equations is a fundamental skill in algebra, and with practice, you'll become more and more proficient. If you have any questions, feel free to review the steps, or try solving similar equations. Keep practicing, stay organized, and always double-check your work. You've got this! Keep learning and exploring the world of math. Math is awesome, and the more you practice, the easier it gets. And the more fun you will have solving mathematical problems. So, what are you waiting for? Start practicing and honing your skills. Happy calculating!