Solving Math Problems: Step-by-Step Guide

Hey guys, let's break down this math problem step-by-step! It's all about understanding the order of operations and being careful with those positive and negative signs. We're going to solve the equation -3(3x - 5) + 7(2x - 1). Don't worry, it might look a little intimidating at first, but I promise, it's totally manageable. Just follow along, and you'll be acing these problems in no time. This is a classic example of an algebraic equation where we need to distribute, combine like terms, and isolate the variable, which in this case is 'x'. So, grab your pens and paper, and let's get started on this mathematical journey. Remember, the key here is to stay organized and methodical. Let's dive into how to tackle these types of problems effectively and efficiently. We'll cover each step in detail, ensuring you grasp every concept. Ready to unlock the solution and build your confidence in algebra? Let's go!

Step-by-Step Solution

Step 1: Distribute the Numbers

First things first, we need to handle those parentheses by distributing the numbers outside them. This means multiplying each term inside the parentheses by the number outside. So, we start with -3(3x - 5). Multiply -3 by 3x, which gives us -9x. Then, multiply -3 by -5. Remember, a negative times a negative is a positive, so -3 times -5 equals +15. Our first part of the equation now becomes -9x + 15. Next, we tackle 7(2x - 1). Multiply 7 by 2x, resulting in 14x. Then, multiply 7 by -1, which gives us -7. So, the second part of the equation is now 14x - 7. Now, our equation looks like this: -9x + 15 + 14x - 7. The distribution step is super crucial because it removes the parentheses, which simplifies the equation and makes it easier to work with. It’s like removing a layer of complexity so we can see the core components of the problem clearly. This initial step sets the stage for combining like terms, the next important stage in simplifying the equation. Remember, consistency in applying the distributive property is key to preventing errors and obtaining the correct solution. Always double-check your multiplication, especially with the negative signs.

Step 2: Combine Like Terms

Next up, let's combine the like terms. This means grouping the terms that have the same variable (in this case, 'x') and the constant numbers together. We have -9x and 14x. Combining these gives us -9x + 14x = 5x. Then we have the constants +15 and -7. Combining these gives us 15 - 7 = 8. Our equation is now simplified to 5x + 8. Combining like terms simplifies the equation dramatically. It reduces the number of terms we need to manage and gets us closer to isolating 'x.' Think of it as streamlining the equation so that it can be solved more easily. By collecting similar elements, you create a clearer structure for isolating your variable. This step helps to bring together the pieces, allowing us to focus on the final steps to find the answer. Always be careful with the signs; make sure you are adding or subtracting the terms correctly. This is a crucial part of the solving process, helping you progress toward the ultimate goal: finding the value of ‘x’.

Step 3: Isolate the Variable (x)

Now, let’s get ‘x’ all by itself. Our equation is 5x + 8 = 0 (assuming the full equation is equal to zero, if not, replace the final result with the other side of the original equation). To isolate 5x, we need to get rid of the +8. We do this by subtracting 8 from both sides of the equation. This keeps the equation balanced. So, 5x + 8 - 8 = 0 - 8. This simplifies to 5x = -8. Now we're one step closer to the solution. To get 'x' completely alone, we need to divide both sides of the equation by 5. This cancels out the 5 on the left side. So, 5x / 5 = -8 / 5. This leaves us with x = -8/5, which can also be written as x = -1.6. Isolating the variable is the final key step toward finding the value of ‘x’. Make sure to apply the inverse operation to both sides to maintain the equation's balance. The goal is to get ‘x’ completely by itself on one side. This step can be compared to peeling the layers off an onion – systematically removing terms until you have revealed the final answer. Ensure you always apply the same operation to both sides of the equation to maintain the mathematical correctness of the equation. This meticulous approach is essential to arriving at the right answer.

Step 4: The Solution

So, after all the work, we have found the solution! The value of x is -8/5 or -1.6. And there you have it, guys! We've successfully solved the equation. Now that we've walked through the steps, you should be able to solve similar equations. Remember to take it step-by-step, be careful with those signs, and don't hesitate to practice. The more you practice, the easier it becomes. Keep up the great work! Remember, solving these equations is not just about finding the answer, it's about understanding the process and building your problem-solving skills. Every equation you solve adds to your confidence and your ability to tackle more complex math challenges. So keep at it, and you'll see yourself getting better and better with each problem.

Tips for Success

• Practice, practice, practice! The more you practice, the better you'll become at recognizing patterns and solving equations quickly. Try working through several similar problems to solidify your understanding. Don't be afraid to go back and review the steps if you get stuck. Repetition is key! Working through various problems allows you to build muscle memory, making each step more natural. The more you work, the better you will get at spotting the nuances of the equations. This practice provides a strong foundation, and allows you to apply the concepts more efficiently. Remember, every time you solve a problem, you're reinforcing your understanding and skills.
• Pay attention to the signs. Positive and negative signs can make or break your answer. Double-check your calculations and remember the rules: a negative times a negative is a positive, and so on. When multiplying or dividing with negative numbers, always pause and reassess the operation. Mastering this will avoid common calculation mistakes. Being precise about the signs will boost your accuracy and keep you away from simple errors. Make it a habit to review the signs and ensure your answers match.
• Show your work. Don't try to do too much in your head. Writing out each step helps you stay organized and makes it easier to spot any mistakes. Showing your work is especially important if you make a mistake because you can go back and find it. It also helps in understanding where you might be going wrong if you are struggling with the equation. By writing down each step, you create a roadmap that helps you and others follow your thought process. This allows for easy reviewing and learning from your mistakes. It will improve your overall problem-solving approach.
• Use the correct order of operations. Remember PEMDAS/BODMAS (Parentheses/Brackets, Exponents/Orders, Multiplication and Division, Addition and Subtraction). Solve equations in the correct order to ensure the correct answer. The order of operations ensures uniformity and consistency in mathematical results. Following the proper sequence guarantees accuracy and precision. Making the order of operations your habit will help you solve complex equations systematically. Prioritize learning the rules to solve equations accurately.

Final Thoughts

Alright, that's all, folks! You've now conquered the equation -3(3x - 5) + 7(2x - 1) and I hope that this makes sense to you all. Remember, math is all about practice and understanding. Keep at it, and you'll become a pro in no time. Don't worry if you don't get it right away. The more you practice and the more you try, the better you will get. If you have any questions, don't hesitate to ask! Keep up the great work, and I’ll see you next time!