Verifying The Solution N=-2 For The Equation 1.4n + 2 = 2n + 3.2

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In this article, we will delve into the process of verifying the solution to the equation 1.4n + 2 = 2n + 3.2. The proposed solution is n = -2, and our aim is to demonstrate the steps involved in confirming its validity. This process not only reinforces the understanding of algebraic equations but also highlights the importance of accurate substitution and simplification. Mastering the skill of verifying solutions is crucial in mathematics as it ensures the correctness of the final answer and enhances problem-solving abilities. Throughout this article, we will dissect each step of the verification process, making it clear and easy to follow for anyone interested in mathematics, regardless of their current proficiency level. We will also explore the underlying principles of equation solving and how these principles are applied in the verification process. This article serves as a comprehensive guide to understanding and verifying solutions in algebraic equations, ensuring a solid foundation for more advanced mathematical concepts.

The equation we are examining is 1.4n + 2 = 2n + 3.2. This is a linear equation in one variable, 'n'. Linear equations are fundamental in algebra and often represent real-world scenarios. Solving them involves isolating the variable to find its value. In this case, we have a proposed solution of n = -2. This means we believe that when we substitute -2 for 'n' in the equation, both sides will be equal, thus satisfying the equation. The process of verifying this solution is not just about confirming the answer; it's about understanding the structure of the equation and how equality is maintained through algebraic operations. Verification is a critical step in problem-solving, as it helps to identify potential errors and ensures the solution's accuracy. It also reinforces the concept of equality in mathematics, where both sides of an equation must remain balanced. This understanding is vital for tackling more complex algebraic problems and applications in various fields such as physics, engineering, and economics.

The verification process involves substituting the proposed solution, n = -2, into the original equation 1.4n + 2 = 2n + 3.2. Let's break this down step-by-step:

  1. Substitution: Replace 'n' with '-2' on both sides of the equation. This gives us: 1. 4(-2) + 2 = 2(-2) + 3.2. This step is crucial as it sets the stage for simplifying each side of the equation. It's essential to perform the substitution accurately, ensuring that the negative sign is correctly applied. The goal here is to transform the equation into a numerical expression that can be easily simplified. Accurate substitution is the foundation of the verification process, and any error at this stage will lead to an incorrect conclusion. This process highlights the importance of paying close attention to detail in mathematical operations.

  2. Simplification: Next, we perform the multiplication on both sides: 1. 4 * (-2) equals -2.8, and 2 * (-2) equals -4. So the equation becomes: -2.8 + 2 = -4 + 3.2. This step simplifies the equation by reducing the multiplication operations to single numerical values. It's important to remember the rules of multiplication with negative numbers to ensure accuracy. The simplification process makes it easier to compare both sides of the equation and determine if they are equal. This step demonstrates the order of operations in mathematics, where multiplication is performed before addition. Accurate simplification is key to arriving at the correct conclusion in the verification process.

  3. Further Simplification: Now, we perform the addition on both sides: -2.8 + 2 equals -0.8, and -4 + 3.2 also equals -0.8. This results in the equation: -0.8 = -0.8. This step further reduces the equation to its simplest form, making the comparison straightforward. The addition of a positive and a negative number requires careful attention to the sign and magnitude of each number. The goal is to arrive at a single numerical value on each side of the equation, allowing for a direct comparison. This step highlights the importance of understanding the rules of addition with negative numbers and how they apply in algebraic equations. Accurate arithmetic is essential for a successful verification.

After performing the substitution and simplification, we arrive at the equation -0.8 = -0.8. This is the last line of justification in the verification process. It demonstrates that after substituting n = -2 into the original equation, both sides are indeed equal. This equality confirms that n = -2 is a valid solution to the equation 1.4n + 2 = 2n + 3.2. The final line serves as the conclusive evidence that the proposed solution satisfies the equation. It highlights the fundamental principle of equation solving, where the left-hand side must equal the right-hand side when the correct value of the variable is substituted. This step is the culmination of the entire verification process, providing a definitive answer to whether the proposed solution is correct. The clarity and accuracy of this final line are crucial for effective communication of the solution and the verification process.

The equation -0.8 = -0.8 is the correct answer because it shows the equality of both sides of the original equation after substituting the proposed solution. This equality is the hallmark of a valid solution. When we substitute n = -2 into 1.4n + 2 = 2n + 3.2, and simplify both sides, we end up with -0.8 on the left and -0.8 on the right. This means that the value n = -2 satisfies the equation, making it a true statement. The significance of this result lies in its demonstration that the proposed solution maintains the balance of the equation. If the two sides were not equal, it would indicate that n = -2 is not a solution. Therefore, -0.8 = -0.8 is not just a numerical statement; it is the final, conclusive step in verifying the solution to an algebraic equation. It underscores the importance of equality in mathematics and how it is used to validate solutions.

During the verification process, several common mistakes can lead to incorrect conclusions. One frequent error is incorrect substitution. For example, failing to apply the negative sign properly when substituting a negative value for the variable. This can result in an entirely different equation and a false result. Another common mistake is errors in arithmetic, such as miscalculating the product of decimals or incorrectly adding negative numbers. These errors can easily throw off the simplification process and lead to an incorrect final line. Additionally, misunderstanding the order of operations can lead to errors in simplification. For instance, adding numbers before performing multiplication can change the outcome significantly. To avoid these mistakes, it is crucial to double-check each step, pay close attention to signs and decimal points, and adhere strictly to the order of operations (PEMDAS/BODMAS). Practice and careful attention to detail are key to minimizing these errors and ensuring accurate verification.

In conclusion, the solution n = -2 is verified as a solution to the equation 1.4n + 2 = 2n + 3.2. The last line of the justification, -0.8 = -0.8, definitively demonstrates this. The process involves substituting the proposed solution into the original equation, simplifying both sides, and confirming that they are equal. This verification process is a fundamental skill in algebra, ensuring the accuracy of solutions and reinforcing the understanding of algebraic principles. By meticulously following each step and avoiding common mistakes, one can confidently verify solutions to various equations. This skill is not only crucial for academic success in mathematics but also for practical problem-solving in various fields. The ability to verify solutions accurately enhances one's mathematical proficiency and builds a strong foundation for more advanced mathematical concepts. Mastering this process empowers individuals to approach algebraic problems with confidence and precision.