Solving $7(3r-1)-(r+5)=-52$: A Step-by-Step Guide

Hey guys! Today, we're diving deep into solving a linear equation. We've got a fun one here: $7(3r-1)-(r+5)=-52$. Don't worry if it looks a bit intimidating at first. We're going to break it down step by step, so by the end of this guide, you'll be solving equations like a pro! Linear equations are the foundation of algebra, and mastering them opens the door to more complex mathematical concepts. So, let's jump right in and tackle this equation together! We'll cover everything from the basic principles of equation solving to the nitty-gritty details of this specific problem. Think of this as your ultimate guide to conquering equations – no mathematical problem will be too tough for you after this. We’ll start with a review of the order of operations and the distributive property, then move into combining like terms and isolating the variable. Each step will be explained clearly, with plenty of detail, so you can follow along easily. This isn't just about getting the right answer; it's about understanding the process and building your confidence in math. Remember, math isn't about memorization; it's about understanding the underlying concepts and applying them in different situations. So grab your pencil and paper, and let's get started on this exciting mathematical journey! We promise, by the end, you’ll feel like a true equation-solving wizard.

Understanding the Basics

Before we dive into the equation $7(3r-1)-(r+5)=-52$, let's brush up on some fundamental concepts. Solving equations involves isolating the variable – in this case, 'r' – on one side of the equation. To do this, we use inverse operations. Remember the order of operations (PEMDAS/BODMAS): Parentheses/Brackets, Exponents/Orders, Multiplication and Division, and Addition and Subtraction. We'll be using this in reverse to undo the operations and isolate 'r'. The distributive property is also crucial here. It states that a(b + c) = ab + ac. We’ll use this to eliminate parentheses in our equation. Think of it like this: you're sharing the 'a' with both 'b' and 'c' inside the parentheses. Understanding these basics is like having the right tools in your toolbox before starting a home repair project. You wouldn't try to hammer a nail with a screwdriver, right? Similarly, you need these fundamental concepts to effectively solve equations. Each of these operations—addition, subtraction, multiplication, and division—has an inverse operation that undoes it. For example, the inverse of addition is subtraction, and the inverse of multiplication is division. These inverse operations are the keys to unlocking the solution to our equation. By applying the appropriate inverse operation, we can systematically peel away the layers of the equation until we isolate the variable 'r'. Remember, every step we take should be aimed at simplifying the equation and bringing us closer to our goal of isolating 'r'. With a solid grasp of these foundational concepts, we’re ready to tackle the more complex parts of our equation. Let's keep these principles in mind as we move forward.

Step-by-Step Solution of $7(3r-1)-(r+5)=-52$

Okay, let's get our hands dirty with the equation $7(3r-1)-(r+5)=-52$. Our first step is to apply the distributive property. This means multiplying the 7 by both terms inside the first parentheses and dealing with the negative sign in front of the second parentheses. So, 7 times 3r is 21r, and 7 times -1 is -7. For the second part, the negative sign in front of (r+5) acts like a -1 being multiplied through, giving us -r and -5. After distributing, our equation looks like this: 21r - 7 - r - 5 = -52. See? We've already made progress by eliminating those parentheses! The next crucial step is combining like terms. We have two terms with 'r' (21r and -r) and two constant terms (-7 and -5). Combining 21r and -r gives us 20r. Combining -7 and -5 gives us -12. Now our equation is simplified to: 20r - 12 = -52. We're getting closer! Now, we need to isolate the term with 'r'. To do this, we'll add 12 to both sides of the equation. This cancels out the -12 on the left side, leaving us with 20r = -40. Remember, whatever we do to one side of the equation, we must do to the other to maintain balance. Finally, to isolate 'r', we divide both sides of the equation by 20. This gives us r = -2. And there we have it! The solution to our equation is r = -2. Wasn't that satisfying? We took a seemingly complex equation and broke it down into manageable steps. Each step followed logically from the previous one, leading us to the solution. This step-by-step approach is key to solving any equation, no matter how intimidating it may seem at first. Practice this method, and you'll find that equation solving becomes much easier and more intuitive.

Checking Our Solution

Alright, we've found our solution, r = -2, but how do we know if we're right? The best way to be sure is to check our solution by plugging it back into the original equation. This is like double-checking your work – a crucial step in math! So, let's substitute -2 for 'r' in the equation $7(3r-1)-(r+5)=-52$. We get: 7(3(-2)-1)-((-2)+5) = -52. Now, we simplify. Inside the first parentheses, 3 times -2 is -6, so we have 7(-6-1)-((-2)+5) = -52. Simplifying further inside the parentheses, -6-1 is -7, and -2+5 is 3, giving us 7(-7)-(3) = -52. Multiplying 7 by -7, we get -49. So, our equation now looks like this: -49 - 3 = -52. And guess what? -49 minus 3 is indeed -52! So, -52 = -52. This means our solution, r = -2, is correct! Checking your solution is not just a good habit; it's a vital skill. It gives you confidence that you've solved the problem correctly and helps you catch any mistakes you might have made along the way. Think of it as your mathematical safety net. By plugging the solution back into the original equation, you're essentially reversing the steps you took to solve it, ensuring that everything lines up perfectly. This process not only verifies your answer but also deepens your understanding of the equation and the solution process. So, always remember to check your solution – it’s the mark of a true math whiz!

Common Mistakes to Avoid

When solving equations like $7(3r-1)-(r+5)=-52$, there are some common pitfalls that students often fall into. One frequent mistake is messing up the distributive property. For example, forgetting to distribute the 7 to both terms inside the parentheses, or mishandling the negative sign in front of (r+5). Remember, that negative sign is like a -1 that needs to be multiplied through. Another common error is combining unlike terms. You can only combine terms that have the same variable and exponent. For instance, you can combine 21r and -r, but you can't combine 21r and -7. A third mistake is not following the order of operations correctly. Always simplify inside parentheses first, then distribute, combine like terms, and finally isolate the variable. Rushing through these steps can lead to errors. Forgetting to perform the same operation on both sides of the equation is another critical mistake. Remember, an equation is like a balance scale – what you do to one side, you must do to the other to keep it balanced. Finally, not checking your solution is a huge missed opportunity. As we discussed earlier, plugging your solution back into the original equation is the best way to catch mistakes and ensure your answer is correct. Being aware of these common mistakes can help you avoid them. Think of it as having a map of the mathematical minefield – you know where the dangers are, so you can steer clear of them. Math is all about precision and attention to detail. By taking your time, carefully following the steps, and avoiding these common errors, you'll be well on your way to solving equations with confidence and accuracy.

Practice Problems

Now that we've walked through the solution and discussed common mistakes, it's time to put your skills to the test! Practice makes perfect, and the more you practice solving equations, the more confident and proficient you'll become. Here are a few practice problems similar to $7(3r-1)-(r+5)=-52$ for you to try:

1. Solve for x: 5(2x + 3) - (x - 4) = 22
2. Solve for y: 3(4y - 2) + 2(y + 1) = -16
3. Solve for a: 8(a - 1) - 3(2a + 5) = -19

Take your time, follow the steps we discussed, and remember to check your solutions. These problems will help you reinforce your understanding of the distributive property, combining like terms, and isolating the variable. Don't be afraid to make mistakes – they're a natural part of the learning process. The key is to learn from your mistakes and keep practicing. You can even break down each problem into smaller steps, just like we did with our main equation. Start by distributing, then combine like terms, isolate the variable term, and finally, solve for the variable. And, of course, don't forget to check your answer by plugging it back into the original equation. If you get stuck, revisit the steps we outlined earlier or ask for help from a teacher, tutor, or friend. Remember, everyone learns at their own pace, and there's no shame in seeking assistance when you need it. Math is a collaborative endeavor, and working together can often lead to deeper understanding and greater success. So, grab your pencil and paper, and let's tackle these practice problems together! You've got this!

Conclusion

So, we've successfully solved the equation $7(3r-1)-(r+5)=-52$, and hopefully, you feel much more confident in your equation-solving abilities now! We broke down each step, from applying the distributive property to combining like terms and isolating the variable. We also emphasized the importance of checking your solution and avoiding common mistakes. Remember, solving equations is like building a puzzle – each step fits together to reveal the final solution. The key is to approach each problem systematically and with a clear understanding of the underlying principles. Don't rush, take your time, and focus on accuracy. Math can be challenging, but it's also incredibly rewarding. The satisfaction of solving a complex problem is a feeling like no other. And the skills you develop in math – critical thinking, problem-solving, attention to detail – are valuable in all aspects of life. So, keep practicing, keep learning, and keep pushing yourself. The world of math is vast and fascinating, and there's always something new to discover. Whether you're solving equations, exploring geometry, or delving into calculus, the journey is just as important as the destination. Embrace the challenges, celebrate your successes, and never stop asking questions. And remember, math isn't just about numbers and formulas; it's about thinking logically and creatively. So, go forth and conquer those equations – you've got the tools and the knowledge to succeed! And always remember, we're here to support you on your mathematical journey, so don't hesitate to reach out if you need help. Happy solving!