Solve X + (x + 26) = 64: A Step-by-Step Guide

Hey guys! Today, we're diving into a fun math problem that's super common in algebra: solving equations. Specifically, we're going to break down the equation x + (x + 26) = 64. Don't worry if it looks a little intimidating at first. We'll go through each step together, making sure you understand exactly how to find the value of x. Solving equations is a fundamental skill in math, and once you get the hang of it, you'll be able to tackle all sorts of problems. So, grab your pencils, and let's get started!

Understanding the Basics of Algebraic Equations

Before we jump into solving our specific equation, let's quickly review what an algebraic equation actually is. Algebraic equations are mathematical statements that show the equality between two expressions. These expressions can involve numbers, variables (like our x), and mathematical operations such as addition, subtraction, multiplication, and division. The goal when solving an equation is to isolate the variable on one side of the equation. This means we want to get something like x = (some number). This "some number" is the solution to the equation, the value of x that makes the equation true. Think of it like a puzzle where we need to figure out what x needs to be to make both sides balance.

Key to solving equations is the concept of maintaining balance. Imagine the equation as a weighing scale. Both sides need to have the same weight to stay balanced. Whatever operation we perform on one side of the equation, we must perform the exact same operation on the other side to keep the balance intact. This principle is crucial for correctly solving equations. For instance, if we subtract a number from the left side, we must subtract the same number from the right side. If we multiply the left side by a number, we must multiply the right side by the same number, and so on. This might sound a bit abstract now, but it will become clearer as we work through our example problem. We'll see exactly how maintaining balance helps us isolate x and find its value. Remember, the goal is always to simplify the equation step by step until we have x all alone on one side.

Now, let's talk about the different parts of an equation. We have variables, which are the letters representing unknown values (like our x). We have constants, which are just regular numbers. And we have coefficients, which are the numbers multiplied by the variables. Recognizing these components helps us strategize how to solve the equation. For example, in our equation, we have two x terms. Our first step will likely involve combining these like terms to simplify things. Understanding these basics is like having the right tools for the job. Once we know the tools, we can confidently approach any algebraic equation and start solving it. So, with these fundamentals in mind, let's get back to our original problem and see how we can apply these concepts to find the solution.

Step-by-Step Solution to x + (x + 26) = 64

Okay, let's tackle the equation x + (x + 26) = 64 step by step. Our main goal here is to isolate x on one side of the equation. This means we want to manipulate the equation until we have x equals some number. The first thing we'll do is simplify the equation by removing the parentheses. Remember, parentheses often indicate that we need to perform an operation inside them first, but in this case, the parentheses are simply grouping x + 26. So, we can rewrite the equation without the parentheses: x + x + 26 = 64. This immediately makes the equation look a little less intimidating, right?

Now, we can see that we have two x terms on the left side of the equation. These are called like terms, and we can combine them to further simplify things. When we add x and x, we get 2x. So, our equation now becomes 2x + 26 = 64. See how much cleaner the equation looks now? We've gone from something that might have seemed complicated to a much simpler form. Combining like terms is a crucial step in solving many algebraic equations. It helps us reduce the number of terms and makes the equation easier to work with. This is a technique you'll use frequently, so it's good to get comfortable with it.

Next, we want to isolate the 2x term. To do this, we need to get rid of the +26 on the left side. Remember our balancing principle? Whatever we do to one side, we must do to the other. So, to eliminate the +26, we'll subtract 26 from both sides of the equation. This gives us 2x + 26 - 26 = 64 - 26. On the left side, the +26 and -26 cancel each other out, leaving us with just 2x. On the right side, 64 - 26 equals 38. So, our equation is now 2x = 38. We're getting closer to our solution! We've successfully isolated the term with x in it.

Finally, to solve for x, we need to get rid of the 2 that's multiplying it. Again, we'll use our balancing principle. To undo multiplication, we divide. So, we'll divide both sides of the equation by 2. This gives us (2x) / 2 = 38 / 2. On the left side, the 2s cancel out, leaving us with just x. On the right side, 38 / 2 equals 19. So, we have our solution: x = 19. We've done it! We've successfully solved the equation and found the value of x. Pat yourselves on the back, guys! You've just navigated a classic algebra problem.

Checking Your Solution

Alright, we've found that x = 19 is the solution to our equation x + (x + 26) = 64. But how can we be absolutely sure that we're right? That's where checking our solution comes in. It's like a final safety net to catch any mistakes we might have made along the way. To check our solution, we simply substitute the value we found for x back into the original equation. If both sides of the equation are equal after the substitution, then we know our solution is correct.

So, let's plug x = 19 back into x + (x + 26) = 64. This gives us 19 + (19 + 26) = 64. Now, we need to simplify the left side of the equation. First, we perform the operation inside the parentheses: 19 + 26 = 45. So, our equation now looks like 19 + 45 = 64. Next, we add 19 and 45, which equals 64. So, we have 64 = 64. This is a true statement! The left side of the equation equals the right side when we substitute x = 19. This confirms that our solution is indeed correct.

Checking your solution is a really important habit to develop in algebra. It only takes a few extra minutes, but it can save you from making errors and ensure that you get the right answer. It's like double-checking your work before submitting an assignment. Think of it as the final step in the problem-solving process. By substituting your solution back into the original equation, you're essentially verifying that your answer makes sense in the context of the problem. This not only gives you confidence in your solution but also helps solidify your understanding of the equation itself. So, always remember to check your answers, guys! It's a simple step that can make a big difference.

Practice Problems and Further Learning

Okay, guys, we've successfully solved the equation x + (x + 26) = 64, and we've even learned how to check our solution. But the best way to really master algebraic equations is through practice. Just like learning any new skill, the more you do it, the better you'll become. So, let's talk about some ways you can continue practicing and expanding your knowledge in this area.

First off, try working through some similar practice problems. Look for equations that involve combining like terms, using the distributive property (if there are more complex parentheses), and isolating the variable. You can find plenty of practice problems in textbooks, online resources, or even worksheets your teacher might provide. The key is to start with simpler equations and gradually work your way up to more challenging ones. As you solve more problems, you'll start to recognize patterns and develop a better intuition for how to approach different types of equations.

One great strategy is to change the numbers in our original equation and see if you can still solve it. For example, what if the equation was x + (x + 30) = 70? Or 2x + (x + 15) = 45? By varying the numbers, you'll challenge yourself to apply the same problem-solving steps in slightly different contexts. This helps you avoid just memorizing steps and encourages you to really understand the underlying concepts.

Beyond practice problems, there are tons of resources available for further learning. Websites like Khan Academy and Coursera offer free courses and tutorials on algebra. These resources can provide you with more in-depth explanations of concepts, additional examples, and even video lessons. Don't be afraid to explore different resources and find the ones that best suit your learning style. Some people learn best by reading, while others prefer watching videos or working through interactive exercises. The important thing is to find resources that keep you engaged and help you grasp the material.

Another great way to learn is by working with others. Form a study group with your classmates or find a friend who's also interested in math. You can work through problems together, discuss different approaches, and explain concepts to each other. Teaching someone else is a fantastic way to solidify your own understanding. Plus, it's always more fun to learn with friends! So, remember, guys, practice makes perfect. The more you work with algebraic equations, the more confident and skilled you'll become. And with the resources available, there's no limit to how much you can learn. Keep practicing, keep exploring, and keep challenging yourselves, and you'll be solving even the most complex equations in no time!

Conclusion

So, guys, we've reached the end of our journey through solving the equation x + (x + 26) = 64. We've covered a lot of ground, from understanding the basics of algebraic equations to working through a step-by-step solution and even checking our answer. We've learned the importance of combining like terms, maintaining balance in our equations, and isolating the variable. And most importantly, we've seen that even seemingly complex equations can be broken down into manageable steps.

Remember, the key to mastering algebra is understanding the underlying principles and practicing consistently. Don't be afraid to make mistakes – they're a natural part of the learning process. The important thing is to learn from your mistakes and keep pushing forward. Algebra is a foundational skill that will serve you well in many areas of math and even in everyday life. Whether you're calculating a budget, figuring out measurements for a project, or even just solving a puzzle, the problem-solving skills you develop in algebra will be invaluable.

I hope this guide has been helpful and that you're feeling more confident in your ability to solve algebraic equations. Keep practicing, keep exploring, and keep asking questions. Math can be challenging, but it can also be incredibly rewarding. There's a real sense of accomplishment that comes from cracking a tough problem and understanding how things work. So, embrace the challenge, enjoy the process, and never stop learning. You've got this, guys!