Mastering Linear Equations: Solve For 'n' With X=-2

Hey math explorers! Ever wondered why linear equations are such a big deal? Well, they're everywhere, from calculating your weekly budget to understanding how rockets fly! They're the backbone of so much of our world, and mastering them isn't just about getting good grades; it's about sharpening your analytical mind, guys. Today, we're diving deep into a super interesting linear equation problem that challenges us to solve for an unknown coefficient 'n', rather than the usual variable 'x'. We're given a specific solution for 'x' (which is -2) and tasked with finding 'n' and then evaluating a simple expression, n² - 12. This isn't just about plugging in numbers; it's about understanding the flow of algebra, the logic behind each step, and developing the patience to meticulously work through complex expressions. We're going to break down this problem step-by-step, making sure we cover all the crucial techniques like handling fractions, distributing terms carefully, and isolating our target variable. Prepare to feel the satisfaction of conquering what might initially look like a daunting algebraic puzzle. This journey will reinforce not just your equation-solving skills but also your problem-solving mindset – a valuable asset in any area of life. So, grab your virtual pen and paper, and let's get ready to decode this mathematical mystery together!

Deciphering the Equation: Our Starting Point

Alright, team, let's stare down this beast of an equation: (2n + x)/8 - (5x + n)/6 = 2x + 7/3. Before we even think about touching the numbers, it's super important to understand what we're looking at. The original prompt had a slight typo, missing the operator between x/8 and 5x+n/6, but in the context of a linear equation, we're going to assume it's a subtraction sign and that the terms are grouped as (2n + x)/8 and (5x + n)/6. This is a common interpretation in these types of problems to maintain linearity and solveability. If we don't make this assumption, the problem becomes ambiguous or much harder to solve for an integer 'n'.

So, our working equation is: (2n + x)/8 - (5x + n)/6 = 2x + 7/3.

What are our players here? We have x, which is our variable, and n, which is an unknown coefficient – a number that's just chilling out in the equation, waiting to be discovered. The problem tells us that x = -2 is the solution for the equation. This is our golden ticket! Our ultimate goal is to find the value of n and then calculate n² - 12. Notice how this is a linear equation? That means no x², no square roots, no 1/x – just variables to the power of one. This linearity simplifies things a lot, despite the fractions. Don't let those fractions scare you, though! They're just numbers dressed up in a tricky costume, and we've got the tools to strip them down to their bare essentials. Understanding the structure of the equation is the first, often overlooked, but most critical step to success. It's like planning your route before starting a long road trip; you wouldn't just jump in the car and hope for the best, right? Same logic applies here: understand the terrain before you traverse it.

The First Strategic Move: Plugging in Our Known Value for x

Okay, guys, now that we've deciphered our mission, the first and most logical step is to substitute the given value of x into our equation. The problem explicitly states that x = -2 is the solution. This is awesome because it immediately reduces the number of unknowns in our equation, simplifying it from something with 'x' and 'n' to something that only contains 'n'. Substitution is a foundational algebraic skill, and doing it accurately is paramount. A tiny slip-up here, especially with negative numbers, can derail your entire solution. Let's walk through it carefully.

Our equation is: (2n + x)/8 - (5x + n)/6 = 2x + 7/3

Now, wherever you see an x, we're going to replace it with -2. Let's do it term by term to ensure no errors:

• The first term, (2n + x)/8, becomes (2n + (-2))/8, which simplifies to (2n - 2)/8.
• The second term, (5x + n)/6, becomes (5(-2) + n)/6. Here, 5 * (-2) is -10, so this term simplifies to (-10 + n)/6.
• On the right side, the first term 2x becomes 2(-2), which simplifies to -4.
• The 7/3 term remains unchanged as it doesn't involve x.

So, after substituting x = -2, our equation now looks much cleaner: (2n - 2)/8 - (-10 + n)/6 = -4 + 7/3.

See? It's already looking less intimidating, isn't it? We've successfully removed x from the picture, and now we only have n to worry about. This step highlights the importance of precision and attention to detail, especially when dealing with negative signs. Always remember that subtracting a negative number is the same as adding a positive one, and multiplying by a negative number flips the sign. Taking your time here will save you a headache later, trust me. This is your foundation for the rest of the problem, so make it solid!

Clearing the Way: Eliminating Fractions with the LCM

Alright, algebra warriors! Now that we've substituted x, we're staring at an equation that still has those pesky fractions: (2n - 2)/8 - (-10 + n)/6 = -4 + 7/3. While we could do arithmetic with fractions, that's often where mistakes creep in and where things get unnecessarily complicated. Our smartest move here is to eliminate the fractions entirely! How do we do that, you ask? By finding the Least Common Multiple (LCM) of all the denominators and multiplying the entire equation by it.

Our denominators are 8, 6, and 3. Let's find their LCM:

• Multiples of 8: 8, 16, 24, 32...
• Multiples of 6: 6, 12, 18, 24, 30...
• Multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24, 27...

The smallest number that appears in all these lists is 24. So, our LCM is 24. This number is our magic key to ditching those denominators! The beauty of multiplying the entire equation by the LCM is that it maintains the equality – whatever you do to one side, you do to the other, and to every single term within those sides. Let's apply this power move:

24 * [(2n - 2)/8] - 24 * [(-10 + n)/6] = 24 * [-4] + 24 * [7/3]

Now, let's simplify each term:

• 24 * [(2n - 2)/8] becomes 3 * (2n - 2) (since 24/8 = 3)
• 24 * [(-10 + n)/6] becomes 4 * (-10 + n) (since 24/6 = 4)
• 24 * [-4] becomes -96
• 24 * [7/3] becomes 8 * 7, which is 56 (since 24/3 = 8)

So, our equation transforms into this much more friendly, fraction-free version: 3(2n - 2) - 4(-10 + n) = -96 + 56.

Isn't that a breath of fresh air? This step is a game-changer for solving complex equations. It removes a major source of potential errors and makes the subsequent algebraic manipulation significantly easier. Always take your time to calculate the LCM correctly and then meticulously multiply it across every single term. This systematic approach is what separates the math masters from the math strugglers!

The Algebraic Battle: Distributing, Combining, and Isolating

Alright, folks, we've successfully banished the fractions, and now we're left with a much more manageable equation: 3(2n - 2) - 4(-10 + n) = -96 + 56. This is where our fundamental algebra skills truly shine! We need to systematically distribute, combine like terms, and finally, isolate 'n'.

First, let's tackle the distribution. Remember to multiply the number outside the parentheses by every term inside the parentheses:

• For 3(2n - 2): 3 * 2n gives 6n, and 3 * -2 gives -6. So, this becomes 6n - 6.
• For -4(-10 + n): This one requires extra care because of the negative sign outside. -4 * -10 gives +40, and -4 * n gives -4n. So, this becomes +40 - 4n.

Now, let's simplify the right side of the equation: -96 + 56. This simply evaluates to -40.

So, after distribution and simplifying the right side, our equation is now: 6n - 6 + 40 - 4n = -40.

Next up, combining like terms. We want to group all the terms containing n together and all the constant terms (just numbers) together.

• Terms with n: 6n and -4n. When combined, 6n - 4n equals 2n.
• Constant terms: -6 and +40. When combined, -6 + 40 equals +34.

So, the equation simplifies beautifully to: 2n + 34 = -40.

We're so close now! Our final mission in this stage is to isolate 'n'. To do this, we need to move the constant term (+34) to the other side of the equation. We achieve this by performing the opposite operation on both sides. Since 34 is being added, we subtract 34 from both sides:

2n + 34 - 34 = -40 - 34 2n = -74

Boom! We've successfully simplified the equation to 2n = -74. This entire process requires meticulousness; a single sign error or a miscalculated combination can lead you astray. Always double-check your distribution and your arithmetic. This systematic breakdown ensures clarity and reduces the chances of making those frustrating, small mistakes. You're doing great, keep that focus!

The Grand Finale: Calculating n² - 12 and Reflecting on the Solution

Alright, mathletes, we're in the home stretch! We've meticulously navigated through substitution, fraction elimination, distribution, and combining like terms, leading us to 2n = -74. Our final step to find the value of n is straightforward: divide both sides by 2.

2n / 2 = -74 / 2

This gives us our prized value for n: n = -37.

Fantastic! We've found the unknown coefficient. But wait, the problem isn't quite finished yet! We need to calculate the expression n² - 12. Now that we know n = -37, we simply plug this value into the expression.

• First, square n: (-37)². Remember that squaring a negative number always results in a positive number. 37 * 37 = 1369.
• So, (-37)² = 1369.
• Next, subtract 12 from this result: 1369 - 12.

And there you have it! n² - 12 = 1357.

Now, a quick glance at the options provided in the original problem (A) 35, B) 36, C) 37, D) 38, E) 39, reveals a bit of a conundrum: our calculated result of 1357 does not match any of them. This is a crucial point to address! While our mathematical process, based on the most logical interpretation of the problem statement, is sound and verified, the discrepancy suggests that either the original problem statement (the equation itself) or the provided multiple-choice options might contain an error. In such situations, it's vital to trust your process and your calculations. The method we followed is correct for solving this type of linear equation. If you encounter this in an exam, you'd typically note the discrepancy. The important takeaway here is not just the final number, but the reliable and systematic approach to solving complex equations. We've mastered the steps, even if the problem's integrity itself is a bit shaky. The ability to verify your steps and recognize a potential problem flaw is just as important as getting the