Solve For X: A Step-by-Step Guide To The Equation

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Hey math enthusiasts! Today, we're diving into a classic algebra problem: solving for x in the equation 3 + x * [2 + 3 * (5 + 2 * 3)] = 73. Don't worry if it looks a bit intimidating at first; we'll break it down step by step, making it super easy to understand. This is a fundamental concept in algebra, and mastering it will give you a solid foundation for more complex mathematical problems. Get ready to flex those brain muscles! Understanding how to solve for x isn't just about getting the right answer; it's about developing critical thinking skills and problem-solving abilities that are valuable in all aspects of life. This particular equation involves the order of operations, often remembered by the acronym PEMDAS or BODMAS. This stands for Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). We will meticulously follow these rules to ensure we arrive at the correct solution. Let's get started, shall we?

Understanding the Equation and Order of Operations

Before we jump into the calculations, let's make sure we're on the same page. The equation 3 + x * [2 + 3 * (5 + 2 * 3)] = 73 is an algebraic equation. Our goal is to isolate x and find its value. To do this, we'll use the order of operations, which is crucial for solving equations correctly. The order of operations dictates the sequence in which we perform mathematical operations to evaluate an expression. The order is as follows: Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). In our equation, we have brackets, multiplication, and addition. Following the order of operations is like following a recipe – if you mix up the ingredients or the order in which you add them, the final result will be completely different. We start with the innermost set of parentheses and work our way outwards. This methodical approach ensures that we simplify the equation in a logical and organized manner, preventing errors and leading us to the correct value of x. The beauty of mathematics lies in its structured nature, and the order of operations is a prime example of this structure. It provides a set of rules that, when followed, guarantees consistency and accuracy in our calculations. Understanding and applying these rules is essential not only for solving equations but also for comprehending more advanced mathematical concepts. Let's start with the innermost parentheses: (5 + 2 * 3). We need to perform the multiplication first because it takes precedence over addition. So, 2 * 3 = 6. Now, we replace (5 + 2 * 3) with (5 + 6). That gives us (11). The next step is to substitute this value back into the original equation, working our way towards isolating x.

Step-by-Step Solution

Alright, guys, let's get into the nitty-gritty of solving this equation. We'll go step by step, ensuring you understand every move. Remember our equation: 3 + x * [2 + 3 * (5 + 2 * 3)] = 73.

  1. Simplify the innermost parentheses: We've already identified this as (5 + 2 * 3). Following the order of operations (PEMDAS/BODMAS), we multiply before adding. So, 2 * 3 = 6. This simplifies the expression inside the parentheses to (5 + 6). Therefore, (5 + 6) = 11. Now our equation becomes: 3 + x * [2 + 3 * 11] = 73.
  2. Work within the brackets: Now, let's focus on the brackets: [2 + 3 * 11]. Again, we follow PEMDAS/BODMAS. Multiplication comes before addition. So, 3 * 11 = 33. The expression inside the brackets simplifies to [2 + 33], and 2 + 33 = 35. Now, the equation looks like this: 3 + x * 35 = 73.
  3. Isolate the term with x: Our goal is to get x by itself. The term with x is x * 35. To isolate this term, we need to get rid of the 3 on the left side of the equation. We do this by subtracting 3 from both sides of the equation. This ensures that the equation remains balanced. So, 3 + x * 35 - 3 = 73 - 3. This simplifies to x * 35 = 70.
  4. Solve for x: We now have x * 35 = 70. To solve for x, we need to get rid of the 35 that's multiplying x. We do this by dividing both sides of the equation by 35. Therefore, x * 35 / 35 = 70 / 35. This simplifies to x = 2.

Therefore, the solution to the equation is x = 2. We've successfully solved for x! See? It wasn't as hard as it looked initially, right? It's all about following the steps methodically.

Verification of the Solution

Always double-check your work, people! It's always a good idea to verify your answer to make sure it's correct. We'll substitute x = 2 back into the original equation: 3 + x * [2 + 3 * (5 + 2 * 3)] = 73. Replace x with 2: 3 + 2 * [2 + 3 * (5 + 2 * 3)] = 73. Now, let's work through the equation step by step, following the order of operations:

  1. (Innermost parentheses): (5 + 2 * 3) = (5 + 6) = 11
  2. (Brackets): [2 + 3 * 11] = [2 + 33] = 35
  3. (Multiplication): 2 * 35 = 70
  4. (Addition): 3 + 70 = 73

The equation becomes 73 = 73. Since the left side equals the right side, our solution, x = 2, is correct! Verifying the solution is an essential part of the problem-solving process. It confirms the accuracy of our calculations and builds confidence in our understanding of the mathematical principles involved. This step helps us catch any errors we might have made during the solving process. A simple substitution and careful evaluation are all it takes to validate our answer, ensuring we've successfully solved the equation. The entire process of solving and verifying the solution provides a comprehensive understanding of how to approach similar problems.

Tips for Solving Similar Equations

Alright, here are some helpful tips for tackling equations like this one in the future:

  1. Master the Order of Operations: This is fundamental. Make sure you understand PEMDAS/BODMAS inside and out. It's the key to solving equations correctly. Practice is key, and the more you practice, the more familiar you will become with these steps.
  2. Break Down the Problem: Don't try to solve everything at once. Work step by step, simplifying each part of the equation systematically. Breaking down complex problems into smaller, manageable steps makes them easier to solve and reduces the chance of errors.
  3. Be Organized: Write down each step clearly. This helps you track your work and makes it easier to spot any mistakes. A neat and organized approach not only enhances the clarity of the solution but also streamlines the problem-solving process.
  4. Practice Regularly: The more you practice, the better you'll become. Work through different examples to build your confidence and understanding. Regular practice solidifies your understanding of mathematical concepts and improves your problem-solving skills.
  5. Check Your Work: Always verify your solution by substituting it back into the original equation. This is the best way to ensure your answer is correct. Verification is a crucial step in mathematics. Checking your work not only ensures the accuracy of your answers but also reinforces your understanding of the concepts and techniques involved.
  6. Seek Help When Needed: Don't hesitate to ask for help from teachers, tutors, or classmates if you get stuck. Learning from others can provide new insights and perspectives, making challenging concepts easier to grasp. Collaboration fosters a deeper understanding of the subject matter. The ability to articulate your difficulties and to accept and implement feedback are valuable skills for academic and professional success.

By following these tips and practicing regularly, you'll be well on your way to mastering algebraic equations! Keep at it, and you'll find that solving for x becomes second nature. Remember that every problem you solve enhances your critical thinking abilities and strengthens your foundation in mathematics. So keep practicing and never stop learning! We hope this detailed guide has helped you understand how to solve equations involving brackets and the order of operations.