The Solution To The Equation 1.4n + 2 = 2n + 3.2 Is N = -2

Verifying solutions to equations is a fundamental concept in mathematics. When presented with a potential solution, it is crucial to substitute it back into the original equation to ensure it satisfies the equality. In this case, we are given that $n = -2$ is a solution to the equation $1.4n + 2 = 2n + 3.2$. Our goal is to provide a step-by-step justification, culminating in the final line that confirms the solution's validity. This article will delve into the process of verifying this solution, highlighting the importance of each step in reaching the correct conclusion. Understanding the verification process is essential not only for confirming solutions but also for building a solid foundation in algebraic problem-solving. Let's explore how substituting $n = -2$ into the equation leads to the final, conclusive line of justification, ensuring that both sides of the equation are indeed equal.

Verifying the Solution n = -2

To verify that $n = -2$ is a solution to the equation $1.4n + 2 = 2n + 3.2$, we must substitute this value for $n$ in the equation and simplify both sides. This process involves replacing every instance of $n$ with $-2$ and then performing the arithmetic operations to see if both sides of the equation result in the same value. This step-by-step approach ensures accuracy and helps in identifying any potential errors. It's a critical part of solving algebraic equations because it provides concrete proof that the solution we've found actually works. Without this verification, we cannot be completely confident in our answer. Understanding this process is crucial for anyone studying algebra, as it forms the basis for more complex problem-solving techniques. The substitution step is the bridge between finding a potential solution and knowing for sure that it is the correct one. This makes it an indispensable skill in mathematics.

Step 1: Substitute n = -2 into the Equation

The initial step in verifying the solution involves substituting $n = -2$ into the original equation, $1.4n + 2 = 2n + 3.2$. This means replacing every instance of $n$ with the value $-2$. This substitution transforms the equation from a symbolic representation to a numerical one, allowing us to perform arithmetic operations and compare the results on both sides. The equation now looks like this: $1.4(-2) + 2 = 2(-2) + 3.2$. This is a crucial step as it sets the stage for the simplification process. Accuracy in substitution is paramount because any error here will propagate through the rest of the verification. By carefully replacing each $n$ with $-2$, we create a concrete expression that can be evaluated. This step highlights the fundamental principle of algebraic substitution, which is a cornerstone of mathematical problem-solving.

Step 2: Simplify Both Sides of the Equation

After substituting $n = -2$ into the equation, the next crucial step is to simplify both sides independently. This involves performing the multiplication and addition operations in the correct order, adhering to the basic principles of arithmetic. On the left side of the equation, $1.4(-2) + 2$, we first multiply $1.4$ by $-2$, which yields $-2.8$. Then, we add $2$ to $-2.8$, resulting in $-0.8$. On the right side of the equation, $2(-2) + 3.2$, we multiply $2$ by $-2$, which gives us $-4$. Then, we add $3.2$ to $-4$, which also results in $-0.8$. This simplification process is vital because it transforms the equation into its simplest form, making it easier to compare both sides. Each arithmetic operation must be performed accurately to ensure the correct result. By simplifying both sides, we are essentially reducing the equation to its core components, allowing us to see if the two sides are indeed equal. This step underscores the importance of precision in mathematical calculations.

Step 3: Compare the Simplified Sides

Once we have simplified both sides of the equation, the final step in the verification process is to compare the results. After performing the arithmetic operations, we found that the left side simplifies to $-0.8$ and the right side also simplifies to $-0.8$. Comparing these two values, we can clearly see that they are equal. This equality, $-0.8 = -0.8$, is the conclusive line of the justification. It demonstrates that when $n = -2$, both sides of the original equation, $1.4n + 2 = 2n + 3.2$, yield the same value. This confirms that $n = -2$ is indeed a solution to the equation. The comparison step is the ultimate test of the solution's validity. It provides definitive proof that the value we substituted satisfies the equation. This step highlights the fundamental concept of equality in mathematics, showing that both sides of a valid equation must have the same value. This final comparison solidifies our understanding of the solution and the equation itself.

The Last Line of Justification

The last line of the justification is the culmination of the verification process, where the simplified expressions from both sides of the equation are compared to confirm their equality. In this specific case, after substituting $n = -2$ into the equation $1.4n + 2 = 2n + 3.2$ and simplifying both sides, we arrive at the conclusive statement: $-0.8 = -0.8$. This statement serves as the definitive proof that $n = -2$ is a valid solution to the given equation. The last line is not merely a result; it is the final verdict, the affirmation that the solution holds true. It encapsulates the entire process of substitution and simplification, bringing it to a logical and verifiable end. Without this final comparison, the verification would be incomplete. It is the moment where we see, unequivocally, that the left side of the equation equals the right side when the proposed solution is used. This final line is the cornerstone of the verification process, providing the certainty that the solution is correct. It represents the successful completion of the algebraic verification, solidifying the understanding of the equation and its solution.

Choosing the Correct Option

Among the given options, the correct one is B. $-0.8 = -0.8$. This is because, after substituting $n = -2$ into the equation $1.4n + 2 = 2n + 3.2$ and simplifying both sides, we arrive at this equality. Options A, C, and D do not represent the correct simplified form of the equation after substitution. Understanding why option B is the correct answer involves tracing back through the steps of the simplification process. Each step, from substitution to arithmetic operations, leads us to this final equality. The incorrect options highlight the importance of accurate arithmetic and careful simplification. They serve as a reminder that even a small error in calculation can lead to a wrong conclusion. Option B, in contrast, is the direct result of correctly applying the principles of algebra and arithmetic. It demonstrates the power of a systematic approach to problem-solving. Choosing the correct option is not just about picking the right answer; it's about understanding the underlying mathematical principles that make that answer correct. This choice reinforces the validity of our solution and the accuracy of our calculations.

Conclusion

In conclusion, the solution $n = -2$ is indeed verified as a solution to the equation $1.4n + 2 = 2n + 3.2$. The step-by-step justification process, involving substitution and simplification, culminates in the final line: $-0.8 = -0.8$. This conclusive equality demonstrates that when $n = -2$, both sides of the equation have the same value, confirming the solution's validity. The process of verifying solutions is a critical aspect of algebra, reinforcing the understanding of equations and their solutions. Each step, from the initial substitution to the final comparison, plays a vital role in ensuring the accuracy of the result. Understanding and mastering this process is essential for success in mathematics. It not only helps in verifying solutions but also in developing a deeper understanding of algebraic principles. The ability to verify solutions builds confidence in one's problem-solving skills and lays the foundation for tackling more complex mathematical challenges. This meticulous approach to problem-solving ensures that we not only find the right answer but also understand why it is correct.