Solving -3x + 5 = 2x - 6: A Step-by-Step Guide

Hey guys! Today, we're diving into a common algebra problem: solving a linear equation. Don't worry if it looks intimidating at first. We'll break it down step-by-step so it's super easy to follow. Our mission? To solve the equation -3x + 5 = 2x - 6. Let's get started!

1. Understanding the Equation

Before we jump into solving, let's quickly understand what this equation means. In simple terms, an equation is a mathematical statement that shows that two expressions are equal. In our case, the expression on the left side of the equals sign (-3x + 5) is equal to the expression on the right side (2x - 6). Our goal is to find the value of 'x' that makes this statement true. Think of 'x' as a mystery number we need to uncover.

When you first glance at the equation -3x + 5 = 2x - 6, it might seem like a jumble of numbers and letters. But fear not! Let's break it down piece by piece. On the left side, we have -3x, which means -3 multiplied by our unknown 'x'. Then, we're adding 5 to that result. On the right side, we have 2x, meaning 2 times 'x', and we're subtracting 6 from that. The equals sign (=) is the crucial part, telling us that whatever the result of the left side is, it's the same as the result of the right side. Our quest is to find the magical value of 'x' that makes both sides balance perfectly. Imagine it like a seesaw; we need to find the 'x' that keeps it perfectly level. This initial understanding is key, guys, because it sets the stage for the algebraic maneuvers we're about to perform. Think of it as laying the foundation for a sturdy building – without a solid foundation, the rest won't stand. So, take a deep breath, remind yourself that you've got this, and let's move on to the next step in our solving adventure.

2. Moving the 'x' Terms to One Side

The first step in solving for 'x' is to gather all the terms containing 'x' on one side of the equation. It doesn't matter which side you choose, but for this example, let's move them to the left side. To do this, we need to get rid of the '2x' term on the right side. How do we do that? We use the principle of inverse operations. Since we have '+2x' on the right, we'll subtract '2x' from both sides of the equation. Remember, whatever you do to one side, you must do to the other to keep the equation balanced.

Okay, folks, let's get our algebraic hands dirty! We're tackling the task of moving all those 'x' terms to one side of the equation, specifically the left side in this case. Why? Because it helps us isolate 'x' and eventually figure out its value. The core idea here is to manipulate the equation without changing its fundamental truth. Think of it like rearranging furniture in a room – you're changing the layout, but the room itself is still the same. So, we're staring at our equation: -3x + 5 = 2x - 6. Our target is that pesky '2x' term on the right side. It's like an uninvited guest at our party, and we need to politely ask it to leave. How? By using the power of inverse operations! Since we have '+2x' hanging out on the right, we're going to do the opposite: we're going to subtract '2x'. But here's the golden rule of algebra, guys: whatever you do to one side of the equation, you absolutely have to do to the other side. It's like a seesaw; if you take weight off one side, you have to take the same weight off the other to keep it balanced. So, we subtract '2x' from both sides. This gives us a new equation: -3x + 5 - 2x = 2x - 6 - 2x. Now, it might look a little more complicated, but trust me, we're making progress! We've taken a big step towards isolating 'x', and that's what it's all about. The next step is to simplify things, but for now, bask in the glory of successfully maneuvering those 'x' terms. You're doing great!

This gives us: -3x + 5 - 2x = 2x - 6 - 2x

Now, simplify both sides by combining like terms. On the left side, we can combine -3x and -2x, which gives us -5x. So the equation becomes:

-5x + 5 = -6

3. Moving the Constant Terms to the Other Side

Next, we want to isolate the 'x' term further. This means getting rid of the '+5' on the left side. Again, we use the inverse operation. Since we have '+5', we'll subtract '5' from both sides:

-5x + 5 - 5 = -6 - 5

This simplifies to:

-5x = -11

Alright, team, let's shift our focus to the next crucial step: moving those constant terms – the plain ol' numbers without any 'x' attached – to the other side of the equation. Why? Because we're on a mission to isolate 'x', to get it all by its lonesome on one side of the equals sign. Think of it like separating the ingredients in a recipe so you can work with them individually. Right now, we're looking at -5x + 5 = -6. See that '+5' hanging out on the left side? It's got to go! We need to evict it using our trusty tool: inverse operations. Remember, inverse operations are like the mathematical version of opposites. Addition's opposite is subtraction, multiplication's opposite is division, and so on. Since we have '+5', we're going to do the opposite: we're going to subtract '5'. But, and this is a big but, we have to do it to both sides of the equation. I can't stress this enough, guys! It's the golden rule of equation-solving, the key to keeping everything balanced. If we only subtracted '5' from the left side, the equation would become lopsided, like a seesaw with too much weight on one side. So, we subtract '5' from both sides, giving us: -5x + 5 - 5 = -6 - 5. Now, let's take a moment to appreciate what we've done. We've strategically subtracted '5' to eliminate it from the left side, bringing us one step closer to isolating 'x'. The next step is to tidy things up by simplifying, but for now, give yourself a pat on the back for mastering the art of moving constant terms. You're becoming algebraic wizards!

4. Isolating 'x'

We're almost there! Now we have -5x = -11. The last step is to isolate 'x' completely. Right now, 'x' is being multiplied by -5. To undo this multiplication, we'll use the inverse operation: division. We'll divide both sides of the equation by -5:

-5x / -5 = -11 / -5

This simplifies to:

x = 11/5

Fantastic work, everyone! We've reached the final stage of our algebraic journey: isolating 'x' completely! This is where we unveil the mystery number we've been chasing all along. We've navigated through moving terms, using inverse operations, and simplifying, and now we're at the grand finale. Let's recap where we are: we've got -5x = -11. This tells us that '-5' times 'x' equals '-11'. Our mission, should we choose to accept it (and we do!), is to get 'x' all by itself on one side of the equation. It's like freeing a captured hero from a fortress – we need to break down the walls that are holding it back. In this case, the wall is the '-5' that's clinging to 'x' through multiplication. So, how do we break down that wall? You guessed it: with inverse operations! The inverse operation of multiplication is division. If '-5' is being multiplied by 'x', we need to divide to undo that. But remember the golden rule, guys? We can't just divide one side; we have to divide both sides to keep the equation balanced and fair. So, we divide both sides of the equation by '-5'. This gives us: -5x / -5 = -11 / -5. Now, let's pause for a moment and savor the beauty of this step. On the left side, the '-5' in the numerator and the '-5' in the denominator cancel each other out, leaving us with just 'x'. It's like magic! On the right side, we have '-11 / -5', which is a division problem. A negative divided by a negative is a positive, so we end up with a positive fraction. The next step is to simplify, but for now, let's celebrate this victory. We've successfully isolated 'x', and the solution is within our grasp. You're algebraic rockstars!

5. The Solution

So, our solution is x = 11/5. This can also be written as a mixed number (2 1/5) or a decimal (2.2). We've successfully solved the equation! To double-check our answer, we can substitute this value of 'x' back into the original equation and see if both sides are equal. If they are, we know we've got the right answer.

And there you have it, folks! We've conquered the equation -3x + 5 = 2x - 6 and emerged victorious with the solution x = 11/5! But let's not just stop there; let's take a moment to bask in the glory of our accomplishment and understand exactly what we've achieved. Solving an equation isn't just about finding a number; it's about unlocking a hidden truth, about revealing the value that makes a mathematical statement balance perfectly. It's like cracking a code or solving a puzzle, and the feeling of getting it right is oh-so-satisfying! Our solution, x = 11/5, represents the exact value that, when plugged back into the original equation, makes both sides equal. It's the missing piece of the puzzle, the key that unlocks the equation's secret. But how can we be absolutely sure that we've got it right? Well, that's where the magic of verification comes in! To double-check our answer, we can substitute x = 11/5 back into the original equation: -3x + 5 = 2x - 6. This is like putting the key in the lock and seeing if it turns smoothly. If both sides of the equation end up being the same after the substitution, we know we've hit the jackpot! So, we replace 'x' with '11/5' on both sides and carefully perform the calculations. If the left side equals the right side, we can confidently declare victory. If not, it's back to the drawing board to find any errors in our steps. But let's be optimistic and assume we've nailed it! The feeling of verifying our solution and confirming that it's correct is the ultimate reward for our algebraic efforts. It's like the final flourish on a masterpiece, the signature of the artist. So, congratulations, team! You've not only solved the equation, but you've also understood the process and the importance of verification. You're well on your way to becoming equation-solving masters!

6. Conclusion

Solving linear equations is a fundamental skill in algebra. By following these steps, you can tackle even more complex equations. Remember to always double-check your work and practice regularly. You've got this!

Alright, guys, we've reached the grand finale of our equation-solving adventure! We've battled the terms, wrestled with the inverse operations, and emerged victorious with the solution x = 11/5. But before we hang up our algebraic hats and celebrate, let's take a moment to reflect on the journey we've taken and the valuable skills we've gained. Solving linear equations, like the one we tackled today, is more than just a mathematical exercise; it's a fundamental skill that unlocks a whole world of problem-solving possibilities. Think of it like learning the alphabet – once you've mastered the letters, you can string them together to form words, sentences, and even entire stories! Similarly, understanding how to solve linear equations is a building block for more advanced mathematical concepts. It's a skill that will come in handy in various fields, from science and engineering to economics and finance. But here's the best part: the steps we followed today aren't just applicable to this specific equation; they're a roadmap for tackling a wide range of linear equations. The same principles of moving terms, using inverse operations, and simplifying apply, no matter how complicated the equation might seem at first glance. It's like having a universal key that can unlock countless doors! So, what's the key to mastering this skill? Practice, practice, practice! The more you solve equations, the more comfortable and confident you'll become. It's like learning to ride a bike – at first, it might seem wobbly and challenging, but with enough practice, you'll be cruising along effortlessly. And remember, guys, always double-check your work! It's like having a safety net to catch any mistakes. Substituting your solution back into the original equation is a foolproof way to verify that you've got the right answer. So, pat yourselves on the back for a job well done, and keep practicing those equation-solving skills. You're algebraic superheroes in the making!

Remember, practice makes perfect! Keep up the awesome work, and you'll be solving equations like a pro in no time!