Solving 12 = 2x - 1: A Step-by-Step Guide

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Hey guys! Let's dive into solving this simple algebraic equation together. If you've ever felt a little lost when looking at equations, don't worry! We're going to break it down step by step so it's super easy to understand. Today, we are tackling the equation 12 = 2x - 1. This equation might look intimidating at first, but trust me, it’s totally manageable. We’ll walk through each step, explaining the logic and the math behind it, so you can confidently solve similar problems in the future. Whether you’re a student brushing up on your algebra skills or just curious about how to solve equations, this guide is for you. So, grab your pen and paper, and let’s get started! We will focus on isolating the variable 'x' on one side of the equation. Remember, the golden rule of algebra is that whatever you do to one side of the equation, you must do to the other side to keep things balanced. This is crucial for maintaining the equality and arriving at the correct solution. By the end of this guide, you’ll not only know the answer but also understand the process, making you a more confident problem solver. We’ll also touch on why each step is important and how it contributes to finding the value of 'x'. So, let's jump right in and unlock the mystery behind this equation!

Understanding the Basics

Before we jump into solving, let's quickly brush up on some basic algebraic concepts. Equations are mathematical statements that show the equality between two expressions. In our case, we have 12 = 2x - 1, which means the value on the left side (12) is equal to the value on the right side (2x - 1). The goal is to find the value of the unknown variable, which in this case is 'x'. Variables are symbols (usually letters) that represent unknown quantities. They're like placeholders that we need to fill with the correct number to make the equation true. To solve for 'x', we need to isolate it on one side of the equation. This means getting 'x' by itself, with a coefficient of 1. A coefficient is the number multiplied by the variable (in our equation, the coefficient of 'x' is 2). We’ll use inverse operations to undo the operations that are being performed on 'x'. Inverse operations are operations that “undo” each other. For example, addition and subtraction are inverse operations, and multiplication and division are inverse operations. Understanding these basic concepts is crucial for tackling algebraic equations effectively. They provide the foundation upon which we can build more complex problem-solving skills. So, make sure you’re comfortable with these ideas before moving on. With a solid grasp of these basics, you'll find solving equations much less daunting and a whole lot more fun. Now that we’ve got the groundwork laid, let’s roll up our sleeves and dive into the step-by-step solution!

Step 1: Adding 1 to Both Sides

The first step in solving the equation 12 = 2x - 1 is to isolate the term with 'x'. To do this, we need to get rid of the '-1' on the right side of the equation. Remember, our goal is to get 'x' by itself. To cancel out the '-1', we use the inverse operation, which is adding 1. But here’s the golden rule: whatever you do to one side of the equation, you must do to the other side. This keeps the equation balanced and the equality intact. So, we add 1 to both sides of the equation: 12 + 1 = 2x - 1 + 1. On the left side, 12 + 1 equals 13. On the right side, -1 + 1 cancels out, leaving us with just 2x. This simplifies the equation to 13 = 2x. By adding 1 to both sides, we’ve successfully moved closer to isolating 'x'. This is a crucial step because it simplifies the equation and makes it easier to solve. Think of it like peeling away layers of an onion – we’re slowly but surely getting to the core, which is the value of 'x'. This step demonstrates the power of inverse operations in solving equations. By understanding how to use inverse operations, you can manipulate equations to get the variable you're solving for all by itself. So, we’ve taken the first big step! Now, let's move on to the next stage in our equation-solving adventure.

Step 2: Dividing Both Sides by 2

We've made great progress! Our equation now looks like 13 = 2x. Remember, our ultimate goal is to isolate 'x' completely. Right now, 'x' is being multiplied by 2. To undo this multiplication, we need to perform the inverse operation, which is division. Just like before, we must do the same thing to both sides of the equation to maintain balance. So, we divide both sides by 2: 13 / 2 = (2x) / 2. On the left side, 13 divided by 2 is 6.5. On the right side, 2x divided by 2 simplifies to just 'x'. This leaves us with the equation 6.5 = x. We’ve successfully isolated 'x'! This step is the final piece of the puzzle. By dividing both sides by 2, we've uncovered the value of 'x' that makes the original equation true. This highlights the importance of understanding how to use inverse operations to solve for variables. Each step we've taken has brought us closer to the solution, and now we've arrived! By dividing, we effectively “undid” the multiplication, allowing 'x' to stand alone and reveal its value. This is a powerful technique that you can use in many different algebraic situations. Now that we've found the value of 'x', let’s take a moment to recap and check our work to make sure we’ve got it right.

Step 3: Checking the Solution

We've arrived at a solution, x = 6.5, but it's always a good idea to double-check our work. This step ensures that our answer is correct and that we haven't made any mistakes along the way. To check our solution, we substitute the value of 'x' back into the original equation: 12 = 2x - 1. Replace 'x' with 6.5: 12 = 2(6.5) - 1. Now, we perform the calculations. 2 multiplied by 6.5 is 13, so the equation becomes 12 = 13 - 1. Next, we subtract 1 from 13, which gives us 12. So, the equation becomes 12 = 12. The left side of the equation equals the right side! This confirms that our solution, x = 6.5, is correct. Checking our solution is a crucial step in the problem-solving process. It provides us with confidence that our answer is accurate and that we've followed the correct steps. It's like a final exam for our work, ensuring that we've mastered the material. This step also highlights the importance of accuracy in mathematics. Even a small mistake can lead to an incorrect solution, so taking the time to check our work is always worth it. By verifying our solution, we've not only confirmed our answer but also reinforced our understanding of the problem-solving process. Now that we’ve successfully solved and checked our work, let’s recap the entire process.

Recap: Steps to Solve the Equation

Let's take a moment to recap the steps we took to solve the equation 12 = 2x - 1. This will help solidify your understanding of the process and make it easier to tackle similar problems in the future.

  1. Step 1: Adding 1 to Both Sides: We started by adding 1 to both sides of the equation to isolate the term with 'x'. This gave us 12 + 1 = 2x - 1 + 1, which simplified to 13 = 2x.
  2. Step 2: Dividing Both Sides by 2: Next, we divided both sides of the equation by 2 to isolate 'x'. This gave us 13 / 2 = (2x) / 2, which simplified to 6.5 = x.
  3. Step 3: Checking the Solution: Finally, we checked our solution by substituting x = 6.5 back into the original equation. This gave us 12 = 2(6.5) - 1, which simplified to 12 = 12, confirming that our solution was correct.

These three steps are the core of solving this type of linear equation. By understanding the logic behind each step and practicing regularly, you can confidently solve similar problems. Remember, the key is to isolate the variable by using inverse operations and always maintaining balance in the equation. This recap highlights the importance of having a systematic approach to problem-solving. By breaking down a complex problem into smaller, manageable steps, we can make it much easier to solve. Each step builds upon the previous one, leading us closer to the final solution. Now that we’ve recapped the solution, let’s wrap things up with some final thoughts.

Final Thoughts and Tips

So, we've successfully solved the equation 12 = 2x - 1 and found that x = 6.5. You've now seen how to break down an algebraic equation into manageable steps and solve for the unknown variable. Solving equations is a fundamental skill in mathematics, and mastering it opens the door to more advanced concepts. Remember, the key to success is practice. The more you solve equations, the more comfortable and confident you'll become. Here are a few tips to keep in mind as you continue your math journey:

  • Always show your work: Writing down each step helps you keep track of your progress and makes it easier to identify any mistakes.
  • Check your solutions: As we demonstrated, checking your solutions is a crucial step in ensuring accuracy.
  • Don't be afraid to ask for help: If you're struggling with a concept, don't hesitate to reach out to a teacher, tutor, or friend for assistance.
  • Practice regularly: Consistent practice is the key to mastering any mathematical skill.

Solving equations is like learning a new language – it takes time and effort, but the rewards are well worth it. You'll find that the skills you develop in algebra will be valuable in many areas of life, from everyday problem-solving to advanced scientific pursuits. Keep practicing, stay curious, and you'll be solving equations like a pro in no time! I hope this guide has been helpful in clarifying the process of solving this equation. Keep up the great work, and happy solving!