Solving 12x - 10 = 5x + 11: A Step-by-Step Guide

by Henrik Larsen 49 views

Introduction: Unlocking the Secrets of Linear Equations

Hey guys! Today, we're diving deep into the fascinating world of linear equations. Specifically, we're going to break down the equation 12x - 10 = 5x + 11. Don't worry if that looks a little intimidating at first glance. By the end of this guide, you'll be solving these types of equations like a pro. We'll walk through each step, explaining the why behind the how, so you'll not only get the answer but also understand the underlying principles. Linear equations are fundamental in mathematics and have tons of real-world applications, from calculating distances and speeds to modeling financial growth. So, buckle up and let's get started!

In the realm of algebra, linear equations stand as cornerstones, providing the foundation for understanding more complex mathematical concepts. At their core, these equations represent relationships between variables where the graph forms a straight line. Mastering the art of solving linear equations is not merely an academic exercise; it's a crucial skill that permeates various facets of life, from budgeting and financial planning to scientific research and engineering. The equation 12x - 10 = 5x + 11, while seemingly simple, encapsulates the essence of linear equations and serves as an excellent example for illustrating the step-by-step process of finding the unknown variable, x. This journey into solving linear equations is not just about finding a numerical answer; it's about developing a logical and systematic approach to problem-solving, a skill that will undoubtedly prove invaluable in numerous contexts. The ability to manipulate equations, isolate variables, and interpret solutions is a hallmark of mathematical literacy, and this guide aims to empower you with the tools and understanding necessary to confidently tackle linear equations of all kinds.

The beauty of linear equations lies in their predictability and the straightforward methods used to solve them. Unlike more complex equations that might require advanced techniques, linear equations can be solved using a set of relatively simple algebraic manipulations. These manipulations, rooted in the fundamental properties of equality, allow us to isolate the variable and determine its value. The equation 12x - 10 = 5x + 11 presents a classic example of a linear equation with one variable, a type of equation that is encountered frequently in algebra and beyond. By methodically applying the principles of addition, subtraction, multiplication, and division, we can unravel the equation and reveal the value of x that satisfies the equality. This process of solving is not just about memorizing steps; it's about understanding the logic behind each operation and how it contributes to the overall goal of isolating the variable. In this guide, we'll not only provide the steps to solve this specific equation but also delve into the underlying concepts that make these steps valid and effective. Understanding these concepts will equip you with the flexibility to adapt and apply these techniques to a wide range of linear equations, solidifying your understanding and boosting your confidence in your mathematical abilities.

Step-by-Step Solution: Cracking the Code

Alright, let's dive into the nitty-gritty and solve this equation together! Here’s a step-by-step breakdown:

  1. The Goal: Isolate the Variable

    • First things first, remember our mission: to get x all by itself on one side of the equation. This means we need to move all the x terms to one side and all the constant terms (the numbers) to the other side.

  2. Combining Like Terms: Moving the x Terms

    • We have 12x on the left and 5x on the right. To get them on the same side, let's subtract 5x from both sides of the equation. Remember, whatever we do to one side, we must do to the other to keep the equation balanced. This gives us:
      • 12x - 5x - 10 = 5x - 5x + 11
      • Which simplifies to: 7x - 10 = 11

  3. Isolating the x Term: Moving the Constants

    • Now we need to get rid of the -10 on the left side. To do this, we'll add 10 to both sides:
      • 7x - 10 + 10 = 11 + 10
      • This simplifies to: 7x = 21

  4. Solving for x: The Final Step

    • We're almost there! Now we have 7x = 21. To get x by itself, we need to divide both sides by 7:
      • 7x / 7 = 21 / 7
      • This gives us our final answer: x = 3

  5. Verification

    • To ensure our solution is accurate, we will substitute the obtained value of x back into the original equation. If the left-hand side (LHS) equals the right-hand side (RHS), our solution is correct.
    • Original equation: 12x - 10 = 5x + 11
    • Substitute x = 3: 12(3) - 10 = 5(3) + 11
    • Simplify: 36 - 10 = 15 + 11
    • Further simplification: 26 = 26
    • Since LHS = RHS, our solution x = 3 is verified.

Let's break down the step-by-step solution even further. When tackling an equation like 12x - 10 = 5x + 11, the first crucial step is to identify the goal. Our primary objective is to isolate the variable x on one side of the equation. This means we need to strategically manipulate the equation to gather all x terms on one side and all constant terms on the other. This process relies on the fundamental principle of equality: whatever operation you perform on one side of the equation, you must perform the same operation on the other side to maintain balance. Neglecting this principle can lead to incorrect solutions. In the initial stages, combining like terms is essential. We observed 12x on the left side and 5x on the right. To consolidate these terms, we subtracted 5x from both sides, resulting in 7x - 10 = 11. This step effectively reduces the complexity of the equation, bringing us closer to isolating x. The subsequent step involves isolating the x term. We had 7x - 10 = 11, and to eliminate the -10 from the left side, we added 10 to both sides, leading to 7x = 21. This action brings us even closer to our goal, leaving us with a simplified equation that directly relates the x term to a constant. Finally, to solve for x, we divided both sides of the equation by 7, resulting in x = 3. This final step reveals the value of x that satisfies the original equation. However, the process doesn't end here. To ensure the accuracy of our solution, we must verify it by substituting x = 3 back into the original equation. This verification process confirms that our solution is indeed correct and provides a sense of closure to the problem-solving journey. By meticulously following these steps and understanding the underlying principles, you can confidently solve a wide array of linear equations and strengthen your algebraic skills.

The step-by-step solution is not just a rote set of instructions; it's a carefully orchestrated dance of algebraic manipulations, each step building upon the previous one to lead us closer to the solution. The initial goal of isolating the variable x is the guiding star, informing every decision we make along the way. The principle of equality acts as our compass, ensuring that we maintain balance and avoid straying from the correct path. When we combine like terms, we're essentially streamlining the equation, making it easier to handle. Subtracting 5x from both sides in our example wasn't just an arbitrary move; it was a deliberate action to consolidate the x terms and simplify the equation. Similarly, adding 10 to both sides was a strategic decision to isolate the x term, bringing us closer to revealing the value of x. The act of dividing both sides by 7 was the final flourish, the culmination of our efforts, allowing us to unveil the solution: x = 3. However, even after reaching this point, the journey isn't complete without verification. Substituting the solution back into the original equation is a crucial step in ensuring accuracy and building confidence in our answer. This process serves as a double-check, confirming that our manipulations were correct and that our solution indeed satisfies the equation. By understanding the rationale behind each step and the interconnectedness of the operations, you can develop a deeper appreciation for the elegance and precision of algebra. The ability to solve linear equations is not just a matter of following steps; it's about understanding the underlying principles and applying them strategically to navigate the problem-solving process.

Common Mistakes to Avoid: Pitfalls and How to Dodge Them

We all make mistakes, especially when we're learning something new. But knowing the common pitfalls can help you avoid them! Here are a few things to watch out for when solving linear equations:

  1. Forgetting to Distribute: If there's a number multiplied by a parenthesis, like 2(x + 3), you must distribute the 2 to both the x and the 3. So, it becomes 2x + 6. Don't just multiply the first term!

  2. Not Combining Like Terms Correctly: Make sure you only combine terms that have the same variable and exponent. You can't combine 7x and 2, for example.

  3. Incorrectly Applying the Order of Operations: Remember PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). Make sure you're performing operations in the correct order.

  4. Not Doing the Same Thing to Both Sides: This is the golden rule! If you add, subtract, multiply, or divide on one side, you must do it on the other side to keep the equation balanced.

  5. Sign Errors: Pay close attention to negative signs. They can be tricky! Double-check your work to make sure you've handled them correctly.

  6. Skipping Steps: It might be tempting to rush through the solution, but skipping steps can lead to careless errors. Take your time and write out each step clearly.

To further elaborate on common mistakes in solving linear equations, let's delve into the intricacies of each pitfall and how to effectively avoid them. Forgetting to distribute is a frequent oversight, particularly when dealing with expressions involving parentheses. When an equation presents a term multiplied by a parenthesis, such as 2(x + 3), it's imperative to distribute the multiplier to each term within the parenthesis. Failing to do so can lead to a fundamentally flawed equation and an incorrect solution. The correct application of the distributive property transforms 2(x + 3) into 2x + 6, ensuring that the multiplier affects both x and 3. This seemingly small step can have a significant impact on the outcome of the problem, highlighting the importance of meticulous attention to detail.

Another common mistake lies in the incorrect combination of like terms. Like terms are those that share the same variable and exponent. For instance, 7x and 2x are like terms because they both contain the variable x raised to the power of 1. However, 7x and 2 are not like terms because 2 is a constant term without any variable. Attempting to combine unlike terms is a fundamental error that disrupts the algebraic structure of the equation and leads to inaccurate results. Therefore, it's crucial to carefully identify and combine only like terms, maintaining the integrity of the equation and paving the way for a correct solution. The order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), is a cornerstone of mathematical calculations. Deviating from this order can result in significant errors, especially in more complex equations. For instance, if an equation involves both multiplication and addition, the multiplication must be performed before the addition. Neglecting this rule can lead to a drastically different and incorrect answer. Adhering strictly to the order of operations ensures that calculations are performed in the correct sequence, leading to accurate and reliable solutions. The principle of maintaining balance in an equation is paramount. Whatever operation is performed on one side of the equation must be mirrored on the other side to preserve the equality. This fundamental rule underpins all algebraic manipulations and is crucial for solving equations correctly. Whether it's adding, subtracting, multiplying, or dividing, the operation must be applied consistently to both sides. Neglecting this principle can disrupt the balance of the equation and lead to an invalid solution. Sign errors are notoriously tricky and can easily derail even the most careful problem-solver. Negative signs, in particular, require meticulous attention, as they can easily be overlooked or mishandled. A simple sign error can propagate through the equation, leading to an incorrect final answer. Therefore, it's crucial to double-check each step to ensure that signs are handled correctly and that no errors have crept in. Finally, the temptation to skip steps in the interest of speed can be counterproductive. While it may seem efficient to condense the solution process, skipping steps increases the risk of making careless errors. Each step in solving an equation serves a purpose, and skipping one can disrupt the logical flow and lead to mistakes. It's generally advisable to write out each step clearly, ensuring that the solution process is transparent and that errors can be easily identified and corrected. By being mindful of these common mistakes and taking steps to avoid them, you can significantly improve your accuracy and confidence in solving linear equations.

Practice Makes Perfect: Examples to Try

Okay, now it's your turn to shine! Here are a few more examples for you to practice with:

  1. 3x + 7 = 16
  2. 8 - 2x = 4
  3. 4x - 9 = 2x + 5

Try solving these on your own, using the steps we discussed. And remember, don't be afraid to make mistakes – that's how we learn! The key is to understand the process and practice, practice, practice. The more you solve, the more comfortable you'll become with linear equations. Grab a pen and paper, and let's get to it! You've got this!

To further enhance your understanding and proficiency in solving linear equations, let's delve into the importance of practice and the benefits of working through a variety of examples. While understanding the theoretical concepts and the step-by-step solution process is crucial, the true mastery of solving linear equations comes from consistent practice. Engaging with a range of examples allows you to apply the learned concepts in different contexts, reinforcing your understanding and building confidence. The provided examples, such as 3x + 7 = 16, 8 - 2x = 4, and 4x - 9 = 2x + 5, offer a spectrum of challenges, each requiring a slightly different approach. By tackling these examples independently, you'll develop the ability to identify the appropriate steps and execute them effectively. The first example, 3x + 7 = 16, involves isolating the variable x by subtracting 7 from both sides and then dividing by 3. This exercise reinforces the fundamental steps of solving linear equations. The second example, 8 - 2x = 4, introduces a negative coefficient for the x term, requiring careful attention to sign conventions. Solving this equation involves subtracting 8 from both sides and then dividing by -2, highlighting the importance of accurate sign manipulation. The third example, 4x - 9 = 2x + 5, presents a slightly more complex scenario where x terms appear on both sides of the equation. To solve this, you'll need to combine like terms by subtracting 2x from both sides and adding 9 to both sides, ultimately isolating the variable x. By working through these examples, you'll not only hone your algebraic skills but also develop a deeper appreciation for the nuances of solving linear equations.

Furthermore, the act of practicing with examples is not just about finding the correct answers; it's also about developing problem-solving strategies and building resilience in the face of challenges. When you encounter a problem that seems difficult, don't be discouraged. Instead, break it down into smaller, more manageable steps. Review the fundamental principles, identify the key operations, and execute them methodically. If you make a mistake, don't view it as a failure; see it as an opportunity to learn and grow. Analyze your error, understand where you went wrong, and adjust your approach accordingly. This iterative process of solving, analyzing, and correcting is essential for developing true mastery. The more you practice, the more familiar you'll become with the different types of linear equations and the techniques required to solve them. You'll develop a sense of intuition, allowing you to quickly identify the most efficient path to the solution. You'll also build confidence in your abilities, which will empower you to tackle even more complex mathematical problems. Remember, practice is not just about quantity; it's about quality. Focus on understanding the underlying concepts, applying the steps correctly, and analyzing your results. With consistent effort and a dedication to learning, you can achieve mastery in solving linear equations and unlock a world of mathematical possibilities.

Conclusion: You've Got This!

So, there you have it! We've tackled the equation 12x - 10 = 5x + 11, broken down the steps, and discussed common mistakes to avoid. Remember, solving linear equations is a skill that gets better with practice. Don't be afraid to challenge yourself, and always remember the fundamental principles. With a little patience and persistence, you'll be solving these equations in your sleep! Keep up the great work, guys, and happy solving!

In conclusion, mastering the art of solving linear equations is a journey that requires both understanding and practice. We've explored the intricacies of the equation 12x - 10 = 5x + 11, dissecting the step-by-step solution process and highlighting the underlying principles that govern each manipulation. From isolating the variable to combining like terms, each step is a deliberate action aimed at simplifying the equation and revealing the value of the unknown. We've also delved into the common pitfalls that can derail even the most diligent problem-solvers, emphasizing the importance of attention to detail and adherence to fundamental rules. Forgetting to distribute, incorrectly combining like terms, disregarding the order of operations, neglecting the balance of the equation, making sign errors, and skipping steps are all potential traps that can lead to incorrect solutions. By being aware of these pitfalls and actively working to avoid them, you can significantly enhance your accuracy and efficiency in solving linear equations. Moreover, we've stressed the importance of practice as a cornerstone of mastery. Engaging with a variety of examples allows you to apply the learned concepts in diverse contexts, reinforcing your understanding and building confidence. The more you practice, the more comfortable you'll become with the different types of linear equations and the techniques required to solve them.

The journey of learning to solve linear equations is not just about acquiring a set of skills; it's about developing a problem-solving mindset and cultivating a love for mathematics. Linear equations are not isolated mathematical constructs; they are fundamental tools that can be applied to a wide range of real-world scenarios. From calculating distances and speeds to modeling financial growth and analyzing scientific data, linear equations provide a framework for understanding and solving problems in various disciplines. By mastering linear equations, you're not just learning algebra; you're equipping yourself with a powerful tool that will serve you well in countless situations. So, embrace the challenge, persevere through the difficulties, and celebrate your successes along the way. Remember, every mistake is a learning opportunity, and every problem solved is a step closer to mastery. With dedication and persistence, you can conquer linear equations and unlock a world of mathematical possibilities. Keep practicing, keep learning, and never stop exploring the fascinating world of mathematics. You have the potential to achieve great things, and linear equations are just the beginning of your mathematical journey. So, go forth, solve with confidence, and let your mathematical skills shine!