Solving 2x - 3 + 4x = 21 A Step-by-Step Guide

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Are you struggling with algebraic equations? Don't worry, guys! You're not alone. Many students find algebra a bit tricky at first, but with the right approach, it becomes much easier. In this guide, we'll break down the equation 2x - 3 + 4x = 21 step-by-step and find the correct solution. We will delve into the fundamental principles of solving linear equations, ensuring you grasp each concept thoroughly. By understanding these principles, you'll be equipped to tackle similar problems with confidence. We'll also explore common mistakes to avoid, offering tips and tricks to improve your problem-solving skills. So, grab a pen and paper, and let's get started!

Understanding the Basics of Algebraic Equations

Before we jump into solving the equation, let's quickly review the basics. An algebraic equation is a mathematical statement that shows the equality between two expressions. Our goal is to find the value of the unknown variable (in this case, 'x') that makes the equation true. Think of it like balancing a scale – whatever you do on one side, you must also do on the other to maintain the balance. This principle is crucial for solving equations correctly.

In the equation 2x - 3 + 4x = 21, 'x' is the variable we need to isolate. To do this, we'll use inverse operations, which are operations that undo each other. For example, addition and subtraction are inverse operations, as are multiplication and division. We'll apply these operations strategically to simplify the equation and get 'x' by itself. Remember, the key is to keep the equation balanced at all times. Each step we take should bring us closer to isolating 'x' while maintaining the equality of both sides. This methodical approach is the cornerstone of solving algebraic equations effectively. With practice, you'll become adept at recognizing the steps needed to simplify and solve a wide range of equations.

Step-by-Step Solution of 2x - 3 + 4x = 21

Now, let's get down to the nitty-gritty and solve the equation 2x - 3 + 4x = 21. Follow along carefully, and you'll see how each step leads us to the solution.

Step 1: Combine Like Terms

The first thing we want to do is simplify the equation by combining like terms. Like terms are terms that have the same variable raised to the same power. In our equation, 2x and 4x are like terms. So, let's combine them:

2x + 4x = 6x

Now, our equation looks like this:

6x - 3 = 21

Combining like terms is a fundamental step in simplifying algebraic expressions and equations. It helps to reduce the complexity of the problem, making it easier to isolate the variable. By grouping terms with the same variable and constant terms, we create a more manageable equation. This step not only simplifies the equation but also lays the groundwork for the subsequent steps in the solution process. Recognizing and combining like terms is a crucial skill in algebra, enabling you to solve more complex problems efficiently.

Step 2: Isolate the Variable Term

Next, we want to isolate the term with the variable, which is 6x. To do this, we need to get rid of the -3 on the left side of the equation. Remember our balancing scale? To undo the subtraction, we'll add 3 to both sides:

6x - 3 + 3 = 21 + 3

This simplifies to:

6x = 24

Isolating the variable term is a critical step in solving for the unknown. By performing the same operation on both sides of the equation, we maintain the balance and move closer to isolating 'x'. Adding 3 to both sides effectively cancels out the -3 on the left side, leaving us with the variable term alone. This step demonstrates the importance of using inverse operations to undo the operations in the equation. Isolating the variable term sets the stage for the final step, where we will isolate the variable itself.

Step 3: Solve for x

Now, we have 6x = 24. To solve for 'x', we need to undo the multiplication. The inverse operation of multiplication is division, so we'll divide both sides by 6:

6x / 6 = 24 / 6

This gives us:

x = 4

And there you have it! We've found the solution to the equation. Dividing both sides by 6 isolates 'x', giving us the value that satisfies the equation. This final step underscores the power of inverse operations in solving algebraic equations. By systematically undoing the operations, we arrive at the solution. Solving for 'x' is the culmination of the previous steps, and it provides the answer to the problem. This process highlights the logical progression involved in solving algebraic equations, where each step builds upon the previous one to reach the final solution.

The Correct Answer

So, the correct answer is:

C. x = 4

Common Mistakes to Avoid

Algebra can be tricky, and it's easy to make mistakes. Here are a few common pitfalls to watch out for:

  • Forgetting to Distribute: If you have an expression like 2(x + 3), remember to distribute the 2 to both the x and the 3.
  • Combining Unlike Terms: You can only combine terms that have the same variable raised to the same power. For example, you can't combine 2x and 3x².
  • Not Performing the Same Operation on Both Sides: Remember the balancing scale! Whatever you do on one side, you must do on the other.
  • Sign Errors: Pay close attention to signs, especially when adding or subtracting negative numbers.

Avoiding these common mistakes is crucial for accurate problem-solving in algebra. Each mistake can lead to an incorrect answer, so it's important to be mindful of these pitfalls. Forgetting to distribute can alter the equation significantly, while combining unlike terms introduces errors in simplification. The principle of maintaining balance by performing the same operation on both sides is fundamental, and neglecting this can disrupt the equation's equality. Sign errors are particularly common, especially when dealing with negative numbers, and require careful attention. By being aware of these mistakes and practicing diligence, you can improve your accuracy and confidence in solving algebraic equations.

Tips for Mastering Algebraic Equations

Want to become a pro at solving equations? Here are some tips to help you master the art:

  • Practice Regularly: The more you practice, the better you'll become. Do lots of problems, and don't be afraid to make mistakes – that's how you learn!
  • Show Your Work: Writing down each step helps you keep track of your progress and makes it easier to spot mistakes.
  • Check Your Answers: Once you've found a solution, plug it back into the original equation to make sure it works.
  • Understand the Concepts: Don't just memorize steps – understand why you're doing what you're doing.
  • Ask for Help: If you're stuck, don't hesitate to ask a teacher, tutor, or friend for help.

These tips are designed to help you develop a strong foundation in algebra and improve your problem-solving skills. Regular practice is essential for building proficiency, as it reinforces concepts and familiarizes you with different types of problems. Showing your work not only helps you track your progress but also allows you to identify errors more easily. Checking your answers is a crucial step to ensure accuracy and validate your solution. Understanding the underlying concepts is key to long-term retention and the ability to apply your knowledge to new situations. Finally, seeking help when you're struggling is a sign of strength, and it can provide you with the guidance you need to overcome challenges.

Conclusion

Solving equations like 2x - 3 + 4x = 21 might seem daunting at first, but with a clear understanding of the steps involved, it becomes much more manageable. Remember to combine like terms, isolate the variable term, and solve for 'x'. And most importantly, practice makes perfect! Keep at it, guys, and you'll become an algebra whiz in no time. By following these guidelines and practicing consistently, you'll build the skills and confidence needed to tackle even the most challenging algebraic equations. So, keep practicing, and you'll soon find yourself mastering algebra with ease.