Solving (4x - 2) / 3 = (x - 1) / 1 A Step-by-Step Guide

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Hey guys! Today, we're diving into a classic algebraic equation: (4x - 2) / 3 = (x - 1) / 1. This type of problem often pops up in math classes, and mastering it is a key step in your algebra journey. So, let's break it down step by step, making sure you not only get the answer but also understand the process. We’ll cover everything from the initial setup to verifying our solution, ensuring you're confident in tackling similar problems in the future.

Understanding the Equation

Before we jump into solving, let's take a moment to really understand what this equation is telling us. The equation (4x - 2) / 3 = (x - 1) / 1 is a linear equation, meaning it involves a variable (in this case, 'x') raised to the power of 1. Our goal is to find the value of 'x' that makes the equation true. In simpler terms, we want to find the number that, when substituted for 'x', will make both sides of the equation equal. Think of it like a balancing scale – we need to perform operations on both sides to keep the scale balanced until we isolate 'x'. This principle of maintaining balance is fundamental to solving algebraic equations. We're dealing with fractions here, which might seem intimidating at first, but don't worry! We'll tackle them using a straightforward method: eliminating the denominators. This involves multiplying both sides of the equation by a common multiple of the denominators, which simplifies the equation significantly. Understanding this initial setup is crucial because it sets the stage for the rest of the solution. Without a clear grasp of the equation's structure and our goal, it's easy to get lost in the steps. So, take a deep breath, and let's move on to the next step: clearing those fractions!

Step-by-Step Solution

Okay, let's get our hands dirty and solve this equation! We'll go through each step meticulously, so you can follow along easily. Remember, the key is to stay organized and keep the equation balanced.

1. Clearing the Fractions

The first thing we want to do is eliminate those pesky fractions. To do this, we need to find the least common multiple (LCM) of the denominators. In our equation, (4x - 2) / 3 = (x - 1) / 1, the denominators are 3 and 1. The LCM of 3 and 1 is simply 3. Now, we'll multiply both sides of the equation by 3. This is a crucial step because it allows us to get rid of the fractions and work with a simpler equation. When we multiply the left side, the 3 in the numerator will cancel out the 3 in the denominator, leaving us with just 4x - 2. On the right side, we multiply (x - 1) by 3, which gives us 3(x - 1). So, after clearing the fractions, our equation now looks like this: 4x - 2 = 3(x - 1). See how much cleaner that looks? This step is a game-changer because it transforms a potentially intimidating equation into a much more manageable one. By understanding the concept of LCM and applying it correctly, we've set ourselves up for success in the subsequent steps. This is a common technique in algebra, so mastering it will help you solve a wide range of equations. Remember, the goal is always to simplify the equation while maintaining balance, and clearing fractions is a powerful tool for achieving that.

2. Distributing

Now that we've cleared the fractions, the next step is to distribute the 3 on the right side of the equation. Remember our equation is currently 4x - 2 = 3(x - 1). Distribution involves multiplying the 3 by each term inside the parentheses. So, we multiply 3 by 'x' and then 3 by '-1'. This gives us 3 * x = 3x and 3 * -1 = -3. Therefore, the right side of the equation becomes 3x - 3. Our equation now looks like this: 4x - 2 = 3x - 3. Distributing is a fundamental algebraic operation that helps us simplify equations by removing parentheses. It's based on the distributive property, which states that a(b + c) = ab + ac. In our case, we're applying this property in reverse to expand the expression 3(x - 1). This step is crucial because it allows us to combine like terms and isolate the variable 'x'. Without distribution, we'd be stuck with the parentheses, making it difficult to proceed. So, make sure you're comfortable with this operation, as it's used extensively in algebra and beyond. By correctly distributing the 3, we've transformed the equation into a more manageable form, setting the stage for the next step: isolating the variable.

3. Isolating the Variable

Alright, let's get that 'x' all by itself! Our current equation is 4x - 2 = 3x - 3. The goal here is to gather all the 'x' terms on one side of the equation and all the constant terms on the other side. We can do this by adding or subtracting terms from both sides. Let's start by subtracting 3x from both sides. This will eliminate the 'x' term from the right side and move it to the left side. When we subtract 3x from 4x, we get x. So, the left side becomes x - 2. On the right side, 3x - 3x cancels out, leaving us with just -3. Now our equation looks like this: x - 2 = -3. Next, we need to get rid of the -2 on the left side. To do this, we'll add 2 to both sides. When we add 2 to -2, they cancel out, leaving us with just 'x' on the left side. On the right side, -3 + 2 equals -1. So, finally, we have x = -1. This is our solution! Isolating the variable is a cornerstone of solving algebraic equations. It involves strategically adding or subtracting terms to move them around while maintaining the equation's balance. This process might seem like a puzzle at first, but with practice, it becomes second nature. By isolating 'x', we've found the value that makes the equation true. But before we celebrate, let's make sure our solution is correct by verifying it.

4. Verifying the Solution

We've arrived at a potential solution, x = -1, but it's always a good idea to double-check our work. This step is called verification, and it's crucial for ensuring accuracy. To verify our solution, we'll substitute x = -1 back into the original equation, (4x - 2) / 3 = (x - 1) / 1, and see if both sides are equal. Let's start with the left side. Substituting x = -1 gives us (4 * -1 - 2) / 3. Simplifying this, we get (-4 - 2) / 3, which is -6 / 3, and that equals -2. Now, let's look at the right side. Substituting x = -1 gives us (-1 - 1) / 1, which is -2 / 1, and that equals -2. Voila! Both sides of the equation equal -2 when x = -1. This confirms that our solution is correct. Verification is an essential step in problem-solving, not just in math but in many areas of life. It's a way of checking your work and ensuring that your answer is valid. By substituting the solution back into the original equation, we can catch any errors we might have made along the way. This process also reinforces our understanding of the equation and how it works. So, always remember to verify your solutions – it's a small step that can make a big difference!

Conclusion

Awesome! We've successfully solved the equation (4x - 2) / 3 = (x - 1) / 1, and found that x = -1. We walked through each step carefully, from clearing the fractions to verifying our solution. Remember, the key to mastering algebra is practice and understanding the underlying principles. So, keep practicing, and you'll become a pro at solving equations in no time! If you have any questions or want to try more problems, feel free to ask. Happy solving!

Key Takeaways:

  • Clearing Fractions: Multiplying both sides of the equation by the least common multiple (LCM) of the denominators.
  • Distributing: Multiplying a term by each term inside parentheses.
  • Isolating the Variable: Gathering all 'x' terms on one side and constant terms on the other side.
  • Verifying the Solution: Substituting the solution back into the original equation to check for accuracy.

By mastering these steps, you'll be well-equipped to tackle a wide variety of algebraic equations. Keep up the great work!