Math Equation Verification: Find The Last Line

by Andrew McMorgan 47 views

Hey guys! Ever get stuck on those pesky math problems where you need to verify if a given solution actually works for an equation? It's a super common stumbling block, especially when dealing with algebraic expressions. Today, we're diving deep into just that scenario, using the equation 1.4n + 2 = 2n + 3.2 and the proposed solution n = -2. Our mission, should we choose to accept it (and we totally should!), is to pinpoint the exact last line of the justification that proves this solution is legit. We'll be breaking down the steps, showing you how to substitute, simplify, and ultimately arrive at that satisfying moment of truth. Get ready to flex those math muscles, because understanding this process is key to acing your algebra tests and feeling confident every time you're asked to verify a solution. So, grab your calculators, a notepad, and let's get started on this mathematical adventure together!

Understanding Equation Verification

So, what exactly does it mean to verify a solution in the world of mathematics, especially with equations like 1.4n + 2 = 2n + 3.2? Think of it like this: an equation is basically a balanced scale. The equals sign (=) is the pivot point, and whatever is on the left side must be exactly equal to whatever is on the right side for the equation to be true. When we're given a potential solution, say n = -2, we're essentially being asked to play detective. We need to substitute this value of 'n' into both sides of the equation and see if the scale stays balanced. If, after plugging in the number and doing all the arithmetic, the left side equals the right side, then our solution is verified – it's a true statement! If they don't match, then that particular value of 'n' is not a solution for that equation. This process is crucial because it confirms that we've found the correct value that makes the equation hold true. It's not just about plugging and chugging; it's about demonstrating, step-by-step, that the proposed value satisfies the conditions of the equation. This verification process is a fundamental skill in algebra, helping to build confidence and accuracy in problem-solving. It's the final stamp of approval that says, "Yes, this works!"

Step-by-Step Verification Process

Alright, let's get down to business and actually verify if n = -2 is the magic number for our equation: 1.4n + 2 = 2n + 3.2. This is where the real fun begins, guys! We're going to take it one step at a time.

Step 1: Substitute 'n' on the Left Side.

First up, we focus on the left side of the equation: 1.4n + 2. We're going to replace every 'n' with our proposed solution, -2. So, it becomes: 1.4 * (-2) + 2.

Now, let's do the multiplication: 1.4 multiplied by -2 is -2.8. Don't forget that negative sign!

Next, we add 2: -2.8 + 2. This gives us -0.8.

So, the left side of the equation, when n = -2, evaluates to -0.8.

Step 2: Substitute 'n' on the Right Side.

Now, let's tackle the right side of the equation: 2n + 3.2. Again, we substitute n = -2: 2 * (-2) + 3.2.

Let's do the multiplication: 2 multiplied by -2 is -4.

Then, we add 3.2: -4 + 3.2. This also gives us -0.8.

So, the right side of the equation, when n = -2, also evaluates to -0.8.

Step 3: Compare Both Sides.

We found that the left side equals -0.8, and the right side also equals -0.8. Since -0.8 is indeed equal to -0.8, our solution n = -2 is verified! We did it!

This step-by-step breakdown shows exactly how we arrived at the conclusion. Each calculation is a building block, leading us to the final confirmation. It’s all about careful substitution and accurate arithmetic. Keep these steps in mind for your next verification challenge!

Identifying the Last Line of Justification

Alright, we've gone through the whole verification process for n = -2 in the equation 1.4n + 2 = 2n + 3.2, and we know that it is a valid solution. But the question asks for the last line of the justification. What does that mean in the context of proving our answer? Well, the justification is essentially the series of steps that show the left side equals the right side. The very last line of that justification is the statement that declares the equality of the simplified expressions from both sides. It's the triumphant moment where we see the final numbers match up.

Let's revisit our calculations:

  • Left Side Simplified: We got -0.8.
  • Right Side Simplified: We also got -0.8.

The justification shows us that after substituting n = -2 and performing all the necessary operations, we arrived at a point where the value of the left side is identical to the value of the right side. The last line of this logical argument is the statement that explicitly shows this final equality. It's the ultimate proof that the equation balances out with the given value of 'n'. This line is what seals the deal and confirms the solution. It's not just about getting the right number; it's about presenting the final, irrefutable mathematical statement that demonstrates the truth of the solution. So, when you see the options, you're looking for the one that represents -0.8 = -0.8.

Analyzing the Options

Now, let's look at the potential answers provided and see which one perfectly fits as the last line of our justification. Remember, the last line is the statement that confirms both sides of the equation are equal after plugging in n = -2.

We calculated that the left side simplifies to -0.8, and the right side also simplifies to -0.8. Therefore, the final, concluding statement of our verification should be -0.8 = -0.8.

Let's break down why the other options aren't the correct last line:

  • A. $0.8=0.8$: This is close, but it's incorrect because our calculations resulted in negative 0.8, not positive 0.8. A single sign error makes this option invalid.

  • C. $3=3$: This might be a correct statement on its own, but it doesn't arise from our specific calculation for the equation 1.4n + 2 = 2n + 3.2 with n = -2. It's not the result of our verification process.

  • D. $-3=-3$: Similar to option C, while this is a true mathematical statement, it's not the outcome of substituting n = -2 into our given equation. It doesn't represent the verification steps we took.

Therefore, the only option that accurately reflects the final, verified equality from our calculations is B. $-0.8=-0.8$. This is the crucial last line that confirms n = -2 is indeed a solution to the equation.

Conclusion: The Verified Solution

So there you have it, math whizzes! We've meticulously worked through the equation 1.4n + 2 = 2n + 3.2, substituting n = -2 into both sides. We saw that the left side simplified to -0.8, and the right side also simplified to -0.8. This direct comparison, where -0.8 equals -0.8, is the core of the justification. It's the ultimate proof that n = -2 is a verified solution. The last line of this mathematical argument, the statement that seals the deal, is -0.8 = -0.8. This confirms that our solution satisfies the equation, keeping the balance perfectly intact. Understanding this verification process is super important for building a strong foundation in algebra. Keep practicing, and you'll be a verification pro in no time!