Solving (x-4)-3=5+3(x+4): A Step-by-Step Guide

Hey guys! Today, we're diving into a common algebra problem that many of you might encounter: solving the equation (x-4)-3=5+3(x+4) for x. Don't worry, it might look intimidating at first, but we'll break it down step by step, making it super easy to understand. Whether you're a student brushing up on your algebra skills or just curious about how these things work, you're in the right place. We'll walk through each step, explaining the logic and math behind it, so you'll not only get the answer but also understand the why behind it. So, grab your pencils and let's get started! We're about to turn this equation from a puzzle into a piece of cake. Let's jump right in and tackle this algebraic challenge together!

1. Understanding the Equation

Before we jump into solving, let's make sure we really understand what we're dealing with. Our main goal here is to find the value of x that makes the equation true. Think of it like a balancing act: the left side of the equation needs to equal the right side. To achieve this, we'll use some algebraic techniques to isolate x on one side. When we talk about solving equations, it's like we are detectives trying to uncover a mystery value, which in this case, is x. Each part of the equation plays a vital role, and understanding these roles is the first step to solving the puzzle. Remember, equations are like stories – they have a beginning, a middle, and an end. The beginning is the equation itself, the middle is our solving process, and the end is the value of x we find. So, with our detective hats on, let’s break down this story and see what clues we can find. By taking this approach, we not only solve the equation but also sharpen our problem-solving skills, which are super valuable in all aspects of life. Now, let’s dive deeper into the equation and start unraveling its secrets!

2. Step-by-Step Solution

Okay, let's get down to business and solve this equation step by step. Here’s how we'll do it:

Step 1: Distribute

First up, we need to deal with the parentheses on the right side of the equation. We have 3 multiplied by (x+4). To get rid of these parentheses, we'll distribute the 3 across both terms inside:

5 + 3(x + 4) becomes 5 + 3x + 12

So, our equation now looks like this:

(x - 4) - 3 = 5 + 3x + 12

Distributing is like sharing the love (or the multiplication, in this case) to everyone inside the parentheses. It ensures that we treat each term fairly and accurately, which is super important in algebra. Without this step, we can't properly simplify and move towards isolating x. It's a fundamental move in the game of equation solving, and mastering it will set you up for success in more complex problems down the road. Think of it as unlocking a door – once we distribute, we open up new possibilities for simplification and solving. Now that we've distributed, the equation is starting to look a bit friendlier, and we're one step closer to our goal!

Step 2: Combine Like Terms

Next, let's simplify each side of the equation by combining like terms. On the left side, we have -4 and -3, which are both constants. We can combine them:

(x - 4) - 3 becomes x - 7

On the right side, we have 5 and 12, also constants. Let's combine them as well:

5 + 3x + 12 becomes 3x + 17

Now, our equation is simplified to:

x - 7 = 3x + 17

Combining like terms is like organizing your toolbox – you group similar tools together to make them easier to find and use. In algebra, this means grouping the constants and the terms with x together. It makes the equation cleaner and simpler, which helps us see the next steps more clearly. This process reduces clutter and highlights the key components of the equation, making it less intimidating and more manageable. Think of it as decluttering your workspace before starting a project; it sets the stage for efficient and accurate work. With the equation simplified, we're in a much better position to isolate x and find our solution. So, let's keep moving forward!

Step 3: Move Variables to One Side

Now, we want to get all the terms with x on one side of the equation. It doesn't matter which side we choose, but let's move them to the right side in this case. To do this, we'll subtract x from both sides:

x - 7 - x = 3x + 17 - x

This simplifies to:

-7 = 2x + 17

Moving variables to one side is like gathering your resources in one place before building something. In this case, we're gathering all the x terms together so we can eventually isolate x and find its value. This step is crucial because it consolidates the variable terms, making it easier to deal with them. It's a bit like organizing your ingredients before you start cooking – you want everything in its place so you can proceed smoothly. By keeping the equation balanced and moving the variables strategically, we’re setting ourselves up for the final steps in solving for x. So, let's keep the momentum going and move closer to our solution!

Step 4: Isolate the Variable Term

We're getting closer! Now, we want to isolate the term with x on the right side. To do this, we'll subtract 17 from both sides:

-7 - 17 = 2x + 17 - 17

This simplifies to:

-24 = 2x

Isolating the variable term is like clearing the stage for the main actor. We're getting rid of any distractions (in this case, the +17) so we can focus solely on the term containing x. This step brings us closer to revealing the value of x by simplifying the equation further. Think of it as setting up the final puzzle piece – we've removed all the surrounding pieces so we can fit the last one into place. By performing this operation on both sides of the equation, we maintain balance and ensure that our solution remains accurate. We're now in the home stretch, with only one step left to fully unveil the mystery of x. Let’s keep going!

Step 5: Solve for x

Finally, to solve for x, we need to get x all by itself. Since x is being multiplied by 2, we'll divide both sides by 2:

-24 / 2 = 2x / 2

This gives us:

-12 = x

So, the solution to the equation is x = -12.

Solving for x is like the grand finale of our algebraic journey! This is the moment where we finally reveal the value we've been searching for. By dividing both sides of the equation by the coefficient of x, we isolate x and uncover its true identity. Think of it as the final act of a magic trick, where the hidden answer is revealed to the audience. Each step we’ve taken has led us to this point, and it’s incredibly satisfying to see the solution come to light. With x now standing alone, we’ve successfully navigated the equation and found our answer. It's a testament to the power of systematic problem-solving and the beauty of algebra. Congrats on reaching the solution!

3. Checking Your Answer

It's always a good idea to check your answer to make sure it's correct. To do this, we'll substitute x = -12 back into the original equation:

((-12) - 4) - 3 = 5 + 3((-12) + 4)

Let's simplify each side:

(-16) - 3 = 5 + 3(-8)
-19 = 5 - 24
-19 = -19

Since both sides are equal, our answer x = -12 is correct!

Checking your answer is like proofreading your work before submitting it – it's a crucial step that ensures accuracy and avoids potential errors. By substituting the value we found for x back into the original equation, we verify that our solution holds true. This process not only confirms our answer but also reinforces our understanding of the equation and the steps we took to solve it. Think of it as double-checking your map to make sure you've reached the right destination. It provides peace of mind and solidifies our confidence in the solution. Plus, it’s a great habit to develop in mathematics and beyond, as it promotes carefulness and attention to detail. So, always remember to check your work – it's the final touch that makes your solution bulletproof!

4. Conclusion

And there you have it! We've successfully solved the equation (x-4)-3=5+3(x+4) and found that x = -12. Remember, the key to solving algebraic equations is to break them down into smaller, manageable steps. Distribute, combine like terms, isolate the variable, and don't forget to check your answer!

Solving equations is like embarking on a puzzle-solving adventure, and each step we take brings us closer to the final answer. We started by understanding the equation, then distributed terms, combined like terms, and strategically moved variables to isolate x. Along the way, we emphasized the importance of keeping the equation balanced and checking our solution to ensure accuracy. Think of it as building a house – each step is a layer of construction, and the final check is the inspection that guarantees structural integrity. By following these steps, we've not only found the value of x but also honed our algebraic skills and problem-solving abilities. Remember, practice makes perfect, so keep tackling those equations and embrace the challenge. You've got this! Keep up the great work, and soon you'll be solving even more complex problems with confidence and ease. Until next time, happy solving!