Solve For X 8x - 2 - 5x = 8 A Step-by-Step Guide

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Solving algebraic equations is a fundamental skill in mathematics, and mastering it opens doors to more complex concepts. In this comprehensive guide, we will delve into the step-by-step solution of the equation 8x - 2 - 5x = 8. We will break down each step, explain the underlying principles, and ensure a clear understanding of how to arrive at the correct answer. This detailed approach will not only help you solve this specific problem but also equip you with the tools to tackle similar algebraic challenges. By the end of this guide, you will have a solid grasp of the techniques involved in solving linear equations, enhancing your problem-solving abilities and mathematical confidence. Let's embark on this journey to unravel the solution and master the art of algebra.

Understanding the Basics of Algebraic Equations

Before diving into the solution, it's essential to grasp the basics of algebraic equations. An algebraic equation is a mathematical statement that asserts the equality of two expressions. These expressions can involve variables (represented by letters like x, y, or z), constants (fixed numerical values), and mathematical operations such as addition, subtraction, multiplication, and division. The goal of solving an equation is to find the value(s) of the variable(s) that make the equation true. This involves isolating the variable on one side of the equation by performing operations that maintain the equality. Understanding these fundamental concepts is crucial for successfully navigating algebraic problems and building a strong foundation in mathematics. Let's explore how these principles apply to our specific equation, 8x - 2 - 5x = 8, and how we can systematically solve it to find the value of x.

Combining Like Terms

In the given equation, 8x - 2 - 5x = 8, our first step involves simplifying the equation by combining like terms. Like terms are terms that contain the same variable raised to the same power. In this case, we have two terms with the variable x: 8x and -5x. Combining these terms involves adding their coefficients (the numbers in front of the variable). So, 8x - 5x equals 3x. This simplification reduces the equation to a more manageable form: 3x - 2 = 8. This step is crucial because it streamlines the equation, making it easier to isolate the variable and solve for x. By combining like terms, we reduce complexity and bring the equation closer to its solution, setting the stage for the next steps in our problem-solving process. Understanding and applying this principle is fundamental to solving various algebraic equations efficiently.

Isolating the Variable

After combining like terms, our equation stands as 3x - 2 = 8. The next crucial step is to isolate the variable x on one side of the equation. This involves getting rid of any terms that are added or subtracted from the variable term. In our case, we have -2 subtracted from 3x. To eliminate this, we perform the inverse operation, which is adding 2 to both sides of the equation. This maintains the balance of the equation, ensuring that we are performing a valid algebraic manipulation. Adding 2 to both sides gives us 3x - 2 + 2 = 8 + 2, which simplifies to 3x = 10. This step is pivotal as it brings us closer to finding the value of x by isolating the variable term. The principle of performing the same operation on both sides of the equation is a cornerstone of algebra, ensuring the equality is preserved throughout the solution process.

Solving for x

Having isolated the variable term, our equation now reads 3x = 10. To finally solve for x, we need to undo the multiplication by 3. We achieve this by performing the inverse operation, which is division. Dividing both sides of the equation by 3 will isolate x and reveal its value. So, we divide both sides by 3: (3x) / 3 = 10 / 3. This simplifies to x = 10/3. This is our solution for x. By dividing both sides of the equation by the coefficient of x, we have successfully found the value of x that satisfies the original equation. This final step demonstrates the power of inverse operations in algebra, allowing us to systematically unravel equations and find solutions. Understanding this process is essential for mastering algebraic problem-solving and building confidence in your mathematical abilities.

Verification of the Solution

After solving for x, it's always a good practice to verify the solution. This involves substituting the value we found for x back into the original equation to ensure it holds true. Our original equation was 8x - 2 - 5x = 8, and we found that x = 10/3. Let's substitute this value into the equation: 8(10/3) - 2 - 5(10/3) = 8. First, we perform the multiplications: 80/3 - 2 - 50/3 = 8. Now, let's combine the fractions: (80/3) - (50/3) - 2 = 8, which simplifies to 30/3 - 2 = 8. Further simplification gives us 10 - 2 = 8, and finally, 8 = 8. Since the equation holds true, our solution x = 10/3 is correct. This verification step is a crucial part of the problem-solving process, as it confirms the accuracy of our solution and helps prevent errors. It reinforces our understanding of the equation and the steps we took to solve it, providing confidence in our mathematical skills.

Step-by-Step Solution

Let's recap the step-by-step solution to the equation 8x - 2 - 5x = 8:

  1. Combine Like Terms:
    • 8x - 5x - 2 = 8
    • 3x - 2 = 8
  2. Isolate the Variable Term:
    • 3x - 2 + 2 = 8 + 2
    • 3x = 10
  3. Solve for x:
    • (3x) / 3 = 10 / 3
    • x = 10/3
  4. Verification:
    • Substitute x = 10/3 into the original equation:
      • 8(10/3) - 2 - 5(10/3) = 8
      • 80/3 - 2 - 50/3 = 8
      • 30/3 - 2 = 8
      • 10 - 2 = 8
      • 8 = 8 (The equation holds true)

By following these steps, we have successfully solved the equation 8x - 2 - 5x = 8 and verified the solution. Each step is grounded in fundamental algebraic principles, ensuring a clear and logical path to the answer. This step-by-step approach not only helps solve this specific problem but also provides a framework for tackling other algebraic equations with confidence.

Common Mistakes to Avoid

When solving algebraic equations, it's crucial to be aware of common mistakes that can lead to incorrect solutions. One frequent error is failing to combine like terms correctly. For instance, in the equation 8x - 2 - 5x = 8, some might mistakenly subtract 5x from -2 instead of combining it with 8x. Another common mistake is not performing the same operation on both sides of the equation. For example, when isolating the variable, one might add 2 only to the left side of the equation 3x - 2 = 8, resulting in an unbalanced equation. Additionally, errors can occur during the arithmetic operations, such as incorrect multiplication or division. It's also important to double-check the signs of the terms, as a simple sign error can change the entire solution. By being mindful of these common pitfalls and carefully reviewing each step, you can significantly improve your accuracy in solving algebraic equations. Practicing with various examples and verifying your solutions are also effective strategies to avoid these mistakes and build a solid understanding of algebraic principles.

Alternative Approaches to Solving the Equation

While the step-by-step method we've outlined is a standard approach to solving linear equations, there are alternative ways to tackle the equation 8x - 2 - 5x = 8. One such method involves rearranging the equation before combining like terms. Instead of directly combining 8x and -5x, you could rewrite the equation as 8x - 5x - 2 = 8. This might provide a clearer visual separation of the terms involving x, making it easier to combine them correctly. Another approach is to use a slightly different order of operations. For example, you could first add 2 to both sides of the original equation, resulting in 8x - 5x = 10, and then combine like terms. While the end result will be the same, exploring these alternative methods can enhance your understanding of the flexibility within algebra and help you develop a more intuitive approach to problem-solving. Each method offers a unique perspective on the equation, and understanding these different pathways can strengthen your mathematical versatility.

Conclusion

In conclusion, solving the equation 8x - 2 - 5x = 8 involves a series of logical steps grounded in algebraic principles. We began by combining like terms to simplify the equation, then isolated the variable x by performing inverse operations, and finally solved for x to find the solution x = 10/3. We also emphasized the importance of verifying the solution to ensure accuracy and discussed common mistakes to avoid. Furthermore, we explored alternative approaches to solving the equation, highlighting the flexibility within algebraic problem-solving. By mastering these techniques, you can confidently tackle a wide range of algebraic equations and build a strong foundation in mathematics. This comprehensive guide serves not only as a solution to this specific problem but also as a valuable resource for developing your algebraic skills and enhancing your mathematical proficiency. Keep practicing, exploring different approaches, and always verifying your solutions to become a more confident and capable problem solver.