Solving Linear Equations Finding X In -x + 3/7 = 2x - 25/7

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In the realm of mathematics, solving equations is a fundamental skill. Linear equations, in particular, form the bedrock of algebra and are encountered in various real-world applications. This article delves into the process of solving a specific linear equation, providing a step-by-step guide and offering insights into the underlying principles. We will dissect the equation −x+37=2x−257-x + \frac{3}{7} = 2x - \frac{25}{7}, meticulously unraveling each step to arrive at the solution for x. Understanding the methodology behind solving such equations equips learners with the tools to tackle more complex mathematical challenges. The journey begins with a clear understanding of the equation's structure and the properties that govern algebraic manipulations. Let's embark on this mathematical exploration, transforming the unknown into a known quantity.

Understanding the Equation

Before diving into the solution, let's first dissect the equation: −x+37=2x−257-x + \frac{3}{7} = 2x - \frac{25}{7}. This is a linear equation in one variable, x. Our goal is to isolate x on one side of the equation to determine its value. To achieve this, we will employ a series of algebraic operations, ensuring that we maintain the equation's balance. Each operation performed on one side must be mirrored on the other side to preserve equality. The equation comprises terms involving x and constant terms, which we will strategically manipulate to isolate x. This process involves combining like terms, adding or subtracting values from both sides, and potentially multiplying or dividing both sides by a constant. Understanding these fundamental principles is crucial for successfully navigating the solution.

Step-by-Step Solution

  1. Combine x terms: Our initial aim is to gather all terms containing x on one side of the equation. To do this, we can add x to both sides of the equation:

    −x+37+x=2x−257+x-x + \frac{3}{7} + x = 2x - \frac{25}{7} + x

    This simplifies to:

    37=3x−257\frac{3}{7} = 3x - \frac{25}{7}

    This step effectively eliminates the x term from the left side, concentrating it on the right side.

  2. Isolate the x term: Next, we want to isolate the term containing x. To achieve this, we add 257\frac{25}{7} to both sides of the equation:

    37+257=3x−257+257\frac{3}{7} + \frac{25}{7} = 3x - \frac{25}{7} + \frac{25}{7}

    This simplifies to:

    287=3x\frac{28}{7} = 3x

    Which further simplifies to:

    4=3x4 = 3x

    This step isolates the x term on the right side, making it easier to solve for x.

  3. Solve for x: Finally, to solve for x, we divide both sides of the equation by 3:

    43=3x3\frac{4}{3} = \frac{3x}{3}

    This gives us:

    x=43x = \frac{4}{3}

    Therefore, the solution to the equation is x=43x = \frac{4}{3}. This is the value of x that satisfies the original equation. To verify this, we can substitute this value back into the original equation and check if both sides are equal. This final step confirms the accuracy of our solution.

Verification

To ensure our solution is correct, we substitute x=43x = \frac{4}{3} back into the original equation:

−x+37=2x−257-x + \frac{3}{7} = 2x - \frac{25}{7}

Substituting x=43x = \frac{4}{3}:

−43+37=2(43)−257-\frac{4}{3} + \frac{3}{7} = 2(\frac{4}{3}) - \frac{25}{7}

Now, we need to simplify both sides of the equation. Let's start with the left side:

−43+37=−4⋅7+3⋅321=−28+921=−1921-\frac{4}{3} + \frac{3}{7} = \frac{-4 \cdot 7 + 3 \cdot 3}{21} = \frac{-28 + 9}{21} = \frac{-19}{21}

Now, let's simplify the right side:

2(43)−257=83−257=8⋅7−25⋅321=56−7521=−19212(\frac{4}{3}) - \frac{25}{7} = \frac{8}{3} - \frac{25}{7} = \frac{8 \cdot 7 - 25 \cdot 3}{21} = \frac{56 - 75}{21} = \frac{-19}{21}

Since both sides of the equation simplify to −1921\frac{-19}{21}, our solution x=43x = \frac{4}{3} is verified.

Common Mistakes to Avoid

When solving linear equations, several common mistakes can lead to incorrect solutions. One frequent error is failing to apply the same operation to both sides of the equation, disrupting the balance and invalidating the result. For instance, adding a number to one side without adding it to the other will lead to an incorrect answer. Another common mistake involves incorrect handling of negative signs. It's crucial to pay close attention to the signs of the terms when performing operations, as a simple sign error can propagate through the solution and result in a wrong answer. Additionally, errors can arise when combining like terms. Ensure that only terms with the same variable and exponent are combined. For example, 3x and 2x can be combined, but 3x and 2x² cannot. Furthermore, when dealing with fractions, it's essential to find a common denominator before adding or subtracting them. Neglecting this step can lead to incorrect calculations. By being mindful of these common pitfalls and practicing diligently, you can enhance your accuracy and confidence in solving linear equations. Always double-check your work and verify your solution to minimize errors.

Importance of Showing Your Work

In mathematics, showing your work is paramount for several reasons. Firstly, it allows you to track your steps and identify any errors you may have made along the way. By meticulously documenting each step, you can easily pinpoint where a mistake occurred and correct it. This is particularly crucial in complex problems where a single error can cascade through the entire solution. Secondly, showing your work facilitates understanding. The process of writing down each step forces you to think through the logic and reasoning behind it. This deeper engagement with the material enhances your comprehension and retention. Furthermore, showing your work enables others to follow your reasoning. Whether it's a teacher grading your work or a colleague reviewing your solution, a clear and well-documented process allows them to understand your approach and provide feedback. This transparency is essential for effective communication and collaboration in mathematics. In addition to these practical benefits, showing your work also cultivates good problem-solving habits. It encourages a systematic and organized approach to tackling mathematical challenges, which is a valuable skill that extends beyond the classroom. By making a habit of showing your work, you not only improve your accuracy but also deepen your understanding and communication skills in mathematics. This practice is a cornerstone of effective mathematical learning and problem-solving.

Practice Problems

To solidify your understanding of solving linear equations, it's essential to practice with a variety of problems. Here are a few additional equations for you to tackle:

  1. 2x+5=112x + 5 = 11
  2. −3x−7=8-3x - 7 = 8
  3. x2+1=4\frac{x}{2} + 1 = 4
  4. 4x−9=2x+34x - 9 = 2x + 3
  5. 5(x+2)=3(x−1)5(x + 2) = 3(x - 1)

Working through these problems will help you hone your skills and build confidence in your ability to solve linear equations. Remember to show your work for each problem, carefully documenting each step you take. This will not only help you identify any errors but also deepen your understanding of the process. After solving each equation, verify your solution by substituting it back into the original equation to ensure that both sides are equal. If you encounter any difficulties, review the steps outlined in this article or seek assistance from a teacher or tutor. Consistent practice is key to mastering the art of solving linear equations. As you work through more problems, you'll develop a stronger intuition for the steps involved and become more efficient in your problem-solving approach. Embrace the challenge and enjoy the satisfaction of successfully unraveling these mathematical puzzles.

Real-World Applications

Linear equations aren't just abstract mathematical concepts; they have a wide range of real-world applications. From calculating the cost of a taxi ride to determining the trajectory of a projectile, linear equations provide a powerful tool for modeling and solving problems in various fields. In finance, they can be used to calculate loan payments, analyze investments, and determine break-even points. In physics, they can describe motion, forces, and energy. In engineering, they are used in designing structures, analyzing circuits, and optimizing processes. Even in everyday life, we encounter linear equations in situations such as converting temperatures, calculating distances, and budgeting expenses. Understanding how to solve linear equations empowers us to make informed decisions and solve practical problems in a variety of contexts. For example, if you're planning a road trip, you can use a linear equation to calculate the distance you can travel based on your car's fuel efficiency and the amount of fuel you have. Or, if you're comparing different cell phone plans, you can use linear equations to determine the most cost-effective option based on your usage patterns. The ability to apply mathematical concepts to real-world situations is a valuable skill that enhances our problem-solving abilities and decision-making capabilities. Linear equations, in particular, serve as a fundamental building block for more advanced mathematical modeling and analysis, making them an essential tool for navigating the complexities of the world around us.

Conclusion

In conclusion, mastering the art of solving linear equations is a cornerstone of mathematical proficiency. This article has provided a comprehensive guide to solving the equation −x+37=2x−257-x + \frac{3}{7} = 2x - \frac{25}{7}, highlighting the step-by-step process and emphasizing the importance of verifying the solution. We've also explored common mistakes to avoid and the significance of showing your work. By understanding the underlying principles and practicing diligently, you can develop the skills and confidence to tackle a wide range of linear equations. Remember that mathematics is not just about memorizing formulas; it's about understanding the logic and reasoning behind them. As you continue your mathematical journey, embrace the challenges and celebrate the successes along the way. The ability to solve linear equations opens doors to a multitude of applications in various fields, empowering you to analyze problems, make informed decisions, and contribute to the world around you. So, keep practicing, keep exploring, and keep expanding your mathematical horizons. The world of mathematics is vast and fascinating, and the skills you acquire along the way will serve you well in all aspects of life.