Solve 6+3x=5(x-1)-3(x-2) Easily

Welcome to the World of Equations!

Hey there, math enthusiasts and curious minds! Ever looked at a bunch of numbers and letters tangled together and thought, "Whoa, what even is that?" Well, you're not alone! Today, we're going to demystify one of those seemingly complex puzzles: a linear equation. Specifically, we're diving headfirst into the equation 6 + 3x = 5(x-1) - 3(x-2). Don't let the length scare you, guys! By the end of this article, you'll not only understand how to solve it, but you'll also feel like a total equation-solving superstar. This isn't just about finding a numerical answer; it's about building a fundamental skill that underpins so much of mathematics and even everyday problem-solving. We’ll break down every single step, from understanding the basics of what makes an equation linear to the final triumphant moment when we unveil the value of 'x'. Our goal here is to make this process as clear, approachable, and yes, even a little fun as possible. We'll use a friendly, conversational tone, steering clear of overly academic jargon, because learning should be an enjoyable journey, not a stressful one. Think of this as your personal guide, walking you through a forest of numbers and operations, helping you spot the path to the solution. Mastering equations like this one is like learning a new language – once you grasp the grammar and vocabulary, you can construct all sorts of powerful sentences, or in this case, solve all sorts of mathematical challenges! So, buckle up, grab a pen and paper if you like, and let's embark on this exciting mathematical adventure together. You're about to gain a truly valuable skill that extends far beyond the textbook, helping you to logically dissect problems and find elegant solutions in various aspects of life. Ready to unlock the mystery of x? Let's do this!

Understanding the Basics: What's a Linear Equation, Anyway?

Alright, before we jump into the nitty-gritty of solving our specific equation, let's chat a bit about what a linear equation actually is. Picture this: you've got a balance scale, right? An equation is essentially a perfectly balanced scale, where whatever is on one side must equal what's on the other. That's what the equals sign (=) tells us. In our equation, 6 + 3x = 5(x-1) - 3(x-2), the left side (6 + 3x) is perfectly balanced with the right side (5(x-1) - 3(x-2)). Now, why "linear"? Great question! An equation is linear if the highest power of its variable (which is x in our case) is just 1. You won't see any x² (x-squared) or x³ (x-cubed) hanging around in a linear equation. It's all straightforward x (or y, or z, or whatever letter they throw at you!). This simplicity is actually a huge advantage, as it means we can use a set of relatively simple, consistent steps to solve them every single time. We're essentially trying to isolate x, meaning we want to get x all by itself on one side of the equals sign, with a number on the other side. Think of it like a treasure hunt where x is the treasure, and all those numbers and operations are the clues and obstacles we need to navigate. What are these obstacles? Well, we have variables (like x), which are unknown values we're trying to find. Then there are constants (like 6, 1, 2), which are just plain old numbers that don't change. And finally, expressions are combinations of variables and constants, joined by mathematical operations (like 5(x-1)). Understanding these basic components is crucial because it helps us know what we're manipulating and why. Linear equations are foundational because they model countless real-world scenarios – from figuring out how much change you'll get back, to calculating fuel efficiency, to determining how long it takes to reach a destination. They're everywhere, guys, and once you master them, you'll start seeing their patterns and applications all around you. So, keep these foundational ideas in mind as we move forward; they are the bedrock upon which our equation-solving skills will be built. This isn't just abstract math; it's a powerful tool for understanding and interacting with the world. Ready to roll up our sleeves and tackle that equation now?

Step-by-Step Breakdown: Solving Our Mystery Equation

Alright, folks, the moment has arrived! Let's dive into solving 6 + 3x = 5(x-1) - 3(x-2) piece by glorious piece. We'll take this slow and steady, like a master chef preparing a gourmet meal, ensuring every ingredient (or, in our case, every step) is perfectly executed. Remember, our ultimate goal is to get x all by its lonesome self on one side of the equals sign. Let's start transforming this equation from a jumbled puzzle into a clear answer!

Step 1: Distribute and Conquer!

Our very first move when faced with parentheses is to distribute. This means multiplying the number outside the parentheses by every single term inside. It's like sharing: everyone inside gets a piece of what's outside! Look at the right side of our equation: 5(x-1) - 3(x-2). We have two distribution tasks here.

First, let's handle 5(x-1):

• 5 * x gives us 5x.
• 5 * -1 gives us -5. So, 5(x-1) transforms into 5x - 5.

Next, let's tackle -3(x-2). Be super careful with the negative sign here, guys! It's a common spot for little slip-ups.

• -3 * x gives us -3x.
• -3 * -2 gives us +6 (remember, a negative times a negative is a positive!). So, -3(x-2) transforms into -3x + 6.

Now, let's rewrite our entire equation with these expanded terms: 6 + 3x = 5x - 5 - 3x + 6

See? Already looking a bit less intimidating, right? We've shed those pesky parentheses and revealed what was hiding inside!

Step 2: Gather 'Round, Like Terms!

With no more parentheses to worry about, our next mission is to combine like terms. This means grouping together all the x terms and all the constant numbers on each side of the equation separately. Think of it like organizing your socks and t-shirts – you don't mix them up!

The left side of our equation, 6 + 3x, already has its like terms separated (a constant and an x-term), so we don't need to do anything there for now.

Now, let's look at the right side: 5x - 5 - 3x + 6. Here, we have two x terms (5x and -3x) and two constant terms (-5 and +6). Let's combine them:

• For the x terms: 5x - 3x = 2x.
• For the constant terms: -5 + 6 = 1.

So, the entire right side simplifies beautifully to 2x + 1.

Now, our equation looks much cleaner: 6 + 3x = 2x + 1

Fantastic work! We're making real progress. Our equation has gone from a sprawling beast to a much more manageable two-sided expression.

Step 3: Bringing X to One Side

Our next big move is to get all the variable terms (x terms) onto one side of the equation. It doesn't really matter which side you choose (left or right), but often it's easier to move the smaller x term to the side with the larger x term to avoid negatives. In our current equation, 6 + 3x = 2x + 1, we have 3x on the left and 2x on the right. Since 2x is smaller than 3x, let's move 2x to the left side.

To move 2x from the right side, we perform the opposite operation. Since it's a positive 2x, we will subtract 2x from both sides of the equation to keep it balanced. Remember, whatever you do to one side, you must do to the other to maintain equality!

6 + 3x - 2x = 2x + 1 - 2x

Now, simplify both sides:

• On the left: 3x - 2x gives us x. So, the left side becomes 6 + x.
• On the right: 2x - 2x cancels out (becomes 0x, or just 0), leaving us with 1.

So, our equation is now: 6 + x = 1

Look at that! We're getting super close to isolating x. It's starting to reveal its true identity!

Step 4: Isolating the X: Almost There!

We've got 6 + x = 1. Now, our final push to get x all by itself is to move all the constant terms to the other side of the equation. Since we have x on the left, we want to move the 6 from the left side to the right side.

Again, we use the opposite operation. Since 6 is being added to x, we will subtract 6 from both sides of the equation:

6 + x - 6 = 1 - 6

Let's simplify:

• On the left: 6 - 6 cancels out (becomes 0), leaving just x.
• On the right: 1 - 6 gives us -5.

And just like that, we have our solution!

Step 5: The Grand Finale: Finding X!

After all that hard work, the grand reveal is here! We've successfully isolated x:

x = -5

Boom! You've found the treasure, guys! The value of x that makes our original equation true is -5. Isn't that satisfying? But wait, there's one more crucial step that truly seals the deal and confirms your triumph.

Step 6: Don't Just Solve It, Prove It!

Solving an equation is awesome, but checking your answer is the mark of a true math pro. It's like proofreading your essay – you catch any little mistakes before you turn it in! To check our solution, we'll substitute x = -5 back into the original equation and see if both sides really do equal each other.

Original equation: 6 + 3x = 5(x-1) - 3(x-2)

Let's substitute x = -5 into the Left-Hand Side (LHS): LHS = 6 + 3(-5) LHS = 6 - 15 LHS = -9

Now, let's substitute x = -5 into the Right-Hand Side (RHS). This one requires a bit more careful calculation, following the order of operations (parentheses first, then multiplication, then addition/subtraction): RHS = 5((-5)-1) - 3((-5)-2) RHS = 5(-6) - 3(-7) RHS = -30 - (-21) Remember, subtracting a negative is the same as adding a positive! (- (-21)) becomes (+ 21). RHS = -30 + 21 RHS = -9

Voila! Since LHS = -9 and RHS = -9, we can confidently say that LHS = RHS. Our solution, x = -5, is absolutely correct! High five, team! You've not only solved a complex-looking linear equation but also validated your work like a seasoned mathematician. This entire process, from distribution to verification, is the bedrock of algebraic problem-solving, and you've just rocked it. Each of these steps, while seemingly distinct, works in harmony to systematically dismantle the equation and reveal its hidden truth. Understanding why we do each step—like maintaining balance or simplifying expressions—is just as important as knowing how to do it. Keep practicing these steps, and you'll be solving equations in your sleep!

Why Bother with Linear Equations? Real-World Magic!

Okay, guys, you've just conquered a pretty chunky linear equation. But I bet some of you might be thinking, "Why did I just spend all that brainpower on x = -5? When am I ever going to use 5(x-1) - 3(x-2) in my daily life?" That's a totally fair question! The truth is, while you might not solve this exact equation every morning with your coffee, the underlying principles of linear equations are everywhere, woven into the fabric of our everyday lives. Seriously! Understanding how to set up and solve them gives you a powerful tool for logical thinking and problem-solving that extends far beyond the classroom.

Think about budgeting, for example. Let's say you get paid an hourly wage, and you also have some fixed expenses, like your phone bill. If you want to know how many hours you need to work (x) to cover your expenses and save a certain amount, guess what? You're setting up a linear equation! Or maybe you're planning a road trip. You know the distance you need to travel and your average speed. To figure out how long it will take (x), you're using a simple distance = rate * time equation, which is a classic linear relationship. Even something as simple as figuring out how many snacks you can buy with a certain amount of money, given the price per snack and perhaps a coupon, can be modeled with a linear equation. These aren't just abstract math problems; they're blueprints for understanding and managing real-world situations! From calculating sales tax or discounts when you're shopping (saving you some serious cash!), to determining how much ingredient A you need to mix with ingredient B to get a desired blend, linear equations are your silent partners in countless practical scenarios. They help engineers design bridges, economists predict market trends, and even personal trainers help clients achieve fitness goals by modeling progress over time. The ability to translate a word problem into an algebraic expression and then solve it means you can quantify unknowns, make informed decisions, and plan effectively. It's about developing a kind of mathematical literacy that empowers you. So, when you practice solving 6 + 3x = 5(x-1) - 3(x-2), you're not just moving numbers around; you're sharpening your critical thinking, logical reasoning, and problem-solving muscles – skills that are incredibly valuable in any career path or life situation. It's the magic of using simple algebra to unlock complex answers, giving you clarity and control in a world full of variables!

Tips and Tricks for Mastering Equations

Alright, awesome equation solvers, you've done the hard work, but the journey to mastery is an ongoing one! While we've tackled one specific equation, the strategies you learned are applicable to a vast array of linear equations. To truly become an equation wizard, here are some super valuable tips and tricks that will help you solidify your skills and tackle any challenge thrown your way. These aren't just one-off suggestions; they're habits that will serve you well in all your mathematical endeavors.

First up: Practice, practice, practice! I know, I know, it sounds cliché, but it's the absolute truth for math. Solving one equation is a great start, but consistency is key. Try solving similar problems, maybe change the numbers a bit, or even search for