Solutions For $4(2n-4)+3=8n-19$ Explained

Diving Deep into the Equation: $4(2n-4)+3=8n-19$

Hey there, math explorers! Today, we're going on an exciting journey to uncover the mystery behind the equation $4(2n-4)+3=8n-19$. Don't let the numbers and letters intimidate you; solving equations like this is a fundamental skill in algebra, opening doors to understanding countless real-world problems. Our primary goal here is to determine how many solutions this specific equation has. Does it have just one clear answer for 'n'? Does it have endless possibilities, where any value of 'n' makes it true? Or is it a bit of a trick, where no value of 'n' can ever satisfy it? Understanding the different types of solutions for a linear equation is crucial, as each type tells a unique story about the relationship between the two sides of the equation. We’ll break down each step, making sure everything is super clear and easy to follow. We’ll start by simplifying both sides of the equation using basic algebraic properties, then we'll try to isolate our variable, 'n'. This process will reveal the true nature of this particular mathematical statement. Whether you're a student brushing up on your skills or just curious about how these things work, you're in the right place to get a clear, friendly, and comprehensive explanation. Let's roll up our sleeves and unravel this equation together, transforming a seemingly complex problem into a straightforward solution discovery. By the end, you'll not only know the answer for this equation but also have a stronger grasp of how to approach similar algebraic challenges, building confidence in your mathematical abilities. Knowing how many solutions an equation possesses is often more insightful than just finding a single numerical answer, as it speaks volumes about the underlying structure and consistency of the mathematical relationship being described. So, let’s get started on this equation-solving adventure and see what $4(2n-4)+3=8n-19$ truly holds for us!

The First Step: Unpacking Parentheses

Our journey to find the solutions for the equation $4(2n-4)+3=8n-19$ begins with tidying up the left side, specifically by addressing those parentheses. The first crucial step in simplifying any algebraic expression that contains parentheses is to apply the distributive property. This property is a fundamental rule in algebra that essentially tells us to multiply the number or term outside the parentheses by every single term inside the parentheses. Think of it like a mail delivery person: they have to deliver mail to every address on their route! In our equation, the 4 outside the (2n-4) needs to be distributed to both 2n and -4. So, let's break it down: we'll multiply 4 by 2n, which gives us 8n. Then, we'll multiply 4 by -4, which results in -16. After applying the distributive property, the expression 4(2n-4) transforms into 8n - 16. This simple yet powerful step makes the equation much easier to work with, as it removes a layer of grouping and brings us closer to combining like terms. Neglecting this step or making an error here is a common pitfall that can derail the entire solving process, leading to incorrect solutions. It's vital to perform this distribution carefully, paying close attention to the signs of the numbers. A common mistake is only multiplying the 4 by 2n and forgetting to multiply it by the -4. Always remember: every term inside the parentheses gets multiplied by the term outside. So now, our equation looks a lot cleaner: 8n - 16 + 3 = 8n - 19. We've successfully taken the first big leap towards discovering the number of solutions this equation holds. This initial simplification is key to revealing the underlying structure of the equation and preparing it for further manipulation. Without correctly distributing, we would be working with a fundamentally different equation, and our final answer regarding its solutions would be inaccurate. Take a moment to appreciate the power of the distributive property; it's truly the key to unlocking many algebraic expressions!

Combining Like Terms and Simplifying

Now that we've successfully distributed the 4 and removed the parentheses from the left side of our equation, $4(2n-4)+3=8n-19$, which transformed it into 8n - 16 + 3 = 8n - 19, our next mission is to combine the like terms. This step is all about tidying up each side of the equation separately before we start moving terms across the equals sign. Think of like terms as items that belong together, similar to sorting laundry or organizing groceries. In algebra, like terms are those that have the exact same variable part (including exponents) or are just constants (numbers without any variables). On the left side, we have 8n, and then we have two constant terms: -16 and +3. Since both -16 and +3 are constants, they are like terms and can be combined. When we combine -16 and +3, we simply perform the arithmetic operation: -16 + 3 equals -13. The 8n term doesn't have any other 'n' terms to combine with on this side, so it stays as 8n. After this combination, the entire left side of our equation simplifies beautifully to 8n - 13. The right side of the equation, 8n - 19, already consists of two unlike terms (a variable term and a constant term), so there's nothing further to combine on that side at this stage. By combining like terms, we effectively make the equation much more concise and manageable, reducing clutter and preparing it for the next phase of isolating the variable. This step is incredibly important because it allows us to see the true structure of the equation and anticipate the path to finding its solutions. It’s like clearing the decks before setting sail; everything needs to be in its proper place. Errors in combining like terms, especially with negative numbers, are common, so it's always a good idea to double-check your arithmetic. Incorrect simplification here would lead to an entirely different equation, and consequently, a wrong determination of the number of solutions. So, with the left side neatly condensed to 8n - 13, our equation now stands as 8n - 13 = 8n - 19. We are now perfectly poised for the grand reveal, moving all variables to one side and constants to the other. This process is bringing us closer and closer to understanding the final nature of the solutions for this equation.

Solving for 'n': The Road to Discovery

Alright, we've done a fantastic job simplifying the equation $4(2n-4)+3=8n-19$ down to a much more manageable form: 8n - 13 = 8n - 19. Now, this is where the real fun of solving for 'n' begins, as we attempt to isolate our variable. The standard approach for solving linear equations is to gather all the terms containing the variable 'n' on one side of the equation and all the constant terms on the other side. Let's start by trying to move all the 'n' terms to the left side. To do this, we need to eliminate the 8n from the right side. The way we achieve this in algebra is by performing the opposite operation: since we have +8n on the right, we will subtract 8n from both sides of the equation. Remember, whatever you do to one side of an equation, you must do to the other side to keep the equation balanced – it's like a perfectly balanced seesaw! So, if we subtract 8n from the left side (8n - 13 - 8n), the 8n terms cancel each other out, leaving us with just -13. Similarly, if we subtract 8n from the right side (8n - 19 - 8n), those 8n terms also cancel each other out, leaving us with just -19. After performing this operation, our equation dramatically changes. We are left with a very interesting statement: -13 = -19. This is the crucial moment where the number of solutions for our initial equation is revealed. This transformation from an equation with variables to a statement solely involving constants is a tell-tale sign that we've reached a pivotal point in our analysis. The goal of