Solve 3(x+2)=2x-4: Step-by-Step Guide

by Pedro Alvarez 38 views

Hey guys! Let's dive into solving a classic first-degree equation. Today, we're tackling the equation 3(x+2) = 2x - 4. Don't worry, it's not as scary as it looks! We'll break it down step by step, making it super easy to understand. First-degree equations, also known as linear equations, are fundamental in algebra. Mastering them is crucial for tackling more complex mathematical problems later on. Think of them as the building blocks of higher-level math. So, whether you're a student brushing up on your algebra skills or just someone who enjoys a good mathematical challenge, this guide is for you. We'll not only solve this specific equation but also discuss the underlying principles and techniques that you can apply to countless other similar problems. This understanding will empower you to confidently approach any first-degree equation that comes your way. We'll start by understanding the basic principles behind solving equations, such as the importance of maintaining balance and performing the same operations on both sides. Then, we'll dive into the specifics of our equation, showing you exactly how to tackle each step. We'll cover distribution, combining like terms, isolating the variable, and finally, finding the solution. Along the way, we'll highlight common mistakes to avoid, ensuring you're on the right track. By the end of this guide, you'll not only have the solution to 3(x+2) = 2x - 4 but also a solid understanding of how to solve first-degree equations in general. So, grab your pen and paper, and let's get started! Remember, math can be fun, especially when you understand the logic behind it. This journey into the world of linear equations will be both educational and empowering, equipping you with the skills you need to conquer mathematical challenges with confidence. Let's unravel the mystery of 3(x+2) = 2x - 4 together!

Understanding the Basics of First-Degree Equations

Before we jump into the equation 3(x+2) = 2x - 4, let's take a moment to understand the fundamental principles behind solving first-degree equations. These equations, at their core, represent a balance. Think of it like a seesaw – what you do to one side, you must do to the other to maintain equilibrium. This principle is the cornerstone of solving any algebraic equation. In a first-degree equation, our goal is to isolate the variable, which is usually represented by 'x'. Isolating the variable means getting it all by itself on one side of the equation. To achieve this, we use inverse operations. Inverse operations are operations that "undo" each other. For example, addition and subtraction are inverse operations, as are multiplication and division. When we encounter an equation, we need to carefully consider the operations being performed on the variable and then apply the appropriate inverse operations to isolate it. Another key concept is combining like terms. Like terms are terms that have the same variable raised to the same power. For example, 3x and 2x are like terms, but 3x and 2x² are not. We can combine like terms by adding or subtracting their coefficients (the numbers in front of the variables). This simplifies the equation and makes it easier to solve. The distributive property is also crucial. It allows us to multiply a number by a sum or difference inside parentheses. For example, in the expression 3(x+2), we use the distributive property to multiply 3 by both x and 2, resulting in 3x + 6. Understanding these basic principles – maintaining balance, using inverse operations, combining like terms, and applying the distributive property – is essential for solving first-degree equations. They form the foundation upon which we'll build our solution to 3(x+2) = 2x - 4. Remember, practice makes perfect. The more you work with these concepts, the more comfortable and confident you'll become in your ability to solve equations. So, let's keep these principles in mind as we move forward and tackle our specific equation. We'll see how these concepts come to life and guide us to the solution.

Step-by-Step Solution to 3(x+2) = 2x - 4

Alright, let's get down to business and solve the equation 3(x+2) = 2x - 4 step by step. Remember those basic principles we just discussed? We're going to put them into action now. The first thing we notice is the parentheses. To get rid of them, we'll use the distributive property. This means we multiply the 3 outside the parentheses by each term inside: 3 * x = 3x and 3 * 2 = 6. So, our equation now looks like this: 3x + 6 = 2x - 4. Great! We've eliminated the parentheses and simplified the equation a bit. Now, our goal is to get all the 'x' terms on one side of the equation and all the constant terms (the numbers without variables) on the other side. To do this, we'll use inverse operations. Let's start by moving the 2x term from the right side to the left side. To do this, we subtract 2x from both sides of the equation (remember, we need to maintain that balance!): 3x + 6 - 2x = 2x - 4 - 2x. This simplifies to x + 6 = -4. See how we subtracted 2x from both sides? This is a crucial step in isolating the variable. Next, we need to move the +6 from the left side to the right side. Again, we use the inverse operation – subtraction. We subtract 6 from both sides: x + 6 - 6 = -4 - 6. This leaves us with x = -10. And there you have it! We've successfully isolated 'x' and found the solution. So, the solution to the equation 3(x+2) = 2x - 4 is x = -10. But wait, we're not quite done yet. It's always a good idea to check our answer to make sure it's correct. To do this, we substitute -10 back into the original equation and see if both sides are equal.

Verifying the Solution

Okay, we've found our solution, x = -10, but let's make absolutely sure it's correct. Verifying the solution is a crucial step in solving any equation. It gives us the confidence that we haven't made any mistakes along the way. To verify, we substitute x = -10 back into the original equation, 3(x+2) = 2x - 4. Let's start with the left side of the equation: 3(x+2). Substituting -10 for x, we get 3(-10 + 2). Inside the parentheses, -10 + 2 equals -8. So, we have 3 * -8, which equals -24. Now, let's look at the right side of the equation: 2x - 4. Substituting -10 for x, we get 2 * -10 - 4. 2 * -10 equals -20, so we have -20 - 4, which also equals -24. Voila! Both sides of the equation equal -24 when we substitute x = -10. This confirms that our solution is indeed correct. Verifying the solution might seem like an extra step, but it's a valuable habit to develop. It helps prevent errors and ensures that you have a solid understanding of the problem-solving process. Think of it as double-checking your work – a practice that's beneficial in all areas of life, not just math! In this case, our verification confirms that x = -10 is the correct solution to the equation 3(x+2) = 2x - 4. We've not only solved the equation but also proven that our solution is accurate. This process reinforces the importance of both finding the solution and confirming its validity. So, remember to always verify your solutions whenever possible. It's a simple step that can make a big difference in your mathematical journey.

Common Mistakes to Avoid

Now that we've successfully solved the equation 3(x+2) = 2x - 4 and verified our solution, let's talk about some common mistakes that people often make when solving first-degree equations. Being aware of these pitfalls can help you avoid them and improve your accuracy. One of the most common mistakes is forgetting to distribute correctly. Remember, when you have a number multiplied by an expression in parentheses, you need to multiply that number by every term inside the parentheses. For example, in the equation 3(x+2), you need to multiply 3 by both x and 2, resulting in 3x + 6. A common error is to multiply only by the first term (3 * x) and forget about the second term (3 * 2). Another frequent mistake is combining unlike terms. Remember, you can only combine terms that have the same variable raised to the same power. For example, you can combine 3x and 2x, but you cannot combine 3x and 2x². Mixing up like and unlike terms can lead to incorrect simplifications and ultimately, the wrong answer. A third common error is not performing the same operation on both sides of the equation. This violates the fundamental principle of maintaining balance. If you add, subtract, multiply, or divide on one side, you must do the same on the other side. Forgetting this rule can throw off the entire equation and lead to an incorrect solution. Sign errors are also a frequent culprit. Pay close attention to positive and negative signs, especially when moving terms from one side of the equation to the other. Remember that when you move a term, you change its sign. For example, if you move a +2x from the right side to the left side, it becomes -2x. Finally, not verifying the solution is a mistake that can easily be avoided. As we discussed earlier, verifying your solution by substituting it back into the original equation is a simple way to catch errors and ensure accuracy. By being mindful of these common mistakes – incorrect distribution, combining unlike terms, not maintaining balance, sign errors, and not verifying the solution – you can significantly improve your ability to solve first-degree equations accurately and confidently.

Practice Problems and Further Learning

So, we've conquered the equation 3(x+2) = 2x - 4, understood the underlying principles, and learned about common mistakes to avoid. Now, it's time to put your newfound skills to the test! Practice is the key to mastering any mathematical concept, and solving first-degree equations is no exception. To help you solidify your understanding, here are a few practice problems you can try:

  1. 5(x - 1) = 3x + 7
  2. 2(2x + 3) = x - 6
  3. 4x - 9 = x + 3

Work through these problems step by step, applying the techniques we've discussed. Remember to distribute carefully, combine like terms accurately, maintain balance by performing the same operations on both sides, and pay close attention to signs. And most importantly, don't forget to verify your solutions! If you encounter any difficulties, revisit the steps we outlined earlier in this guide. Review the principles of maintaining balance, using inverse operations, combining like terms, and applying the distributive property. Understanding these fundamentals is crucial for success. In addition to these practice problems, there are many excellent resources available for further learning about first-degree equations and algebra in general. Online websites like Khan Academy and Purplemath offer comprehensive lessons, practice exercises, and video tutorials. Textbooks and workbooks can also be valuable resources, providing structured lessons and a wide range of practice problems. Don't hesitate to explore these resources and find the learning methods that work best for you. Remember, learning math is a journey, not a destination. The more you practice and explore, the more confident and proficient you'll become. So, keep practicing, keep learning, and most importantly, keep having fun with math! First-degree equations are just the beginning. As you master these fundamentals, you'll open the door to more advanced mathematical concepts and problem-solving skills. Good luck, and happy solving!