Solving Equations A Step-by-Step Guide With Quinda

Hey guys! Let's dive into a math problem today. We're going to break down how Quinda tackles the equation $-4.5x + 3 = 2 - 8.5x$. Math can seem intimidating, but with a clear step-by-step approach, you'll see it's totally manageable. We'll explore the initial move Quinda makes and then figure out the best next step to isolate our variable, 'x'. So, buckle up, and let's get started!

Quinda's First Move Adding $8.5x$

When solving equations, the ultimate goal is to isolate the variable on one side. In our case, we want to get 'x' all by itself on either the left or right side of the equation. Quinda starts with the equation $-4.5x + 3 = 2 - 8.5x$. The logical first move she makes is to add $8.5x$ to both sides. Why? Well, by adding $8.5x$ to both sides, she aims to eliminate the $-8.5x$ term on the right side. Remember, whatever you do to one side of an equation, you must do to the other to maintain balance – it's like a see-saw! So, let's see what happens when we perform this addition. Adding $8.5x$ to both sides gives us: $-4.5x + 3 + 8.5x = 2 - 8.5x + 8.5x$. Now, let's simplify this. On the left side, we combine the 'x' terms: $-4.5x + 8.5x = 4x$. So, the left side becomes $4x + 3$. On the right side, $-8.5x + 8.5x$ cancels out, leaving us with just 2. Our new equation is now $4x + 3 = 2$. See how much simpler it looks already? By adding $8.5x$, Quinda has successfully moved the 'x' terms closer to one side, which is a crucial step in solving for 'x'. This is a classic algebraic technique – moving like terms together. The addition property of equality is what makes this valid, ensuring that the equation remains balanced throughout the process. The key takeaway here is that by strategically adding the right term, we can simplify the equation and pave the way for isolating the variable. Good job, Quinda!

The Next Strategic Step Separating Variables and Constants

Alright, so Quinda has made a solid start, and our equation now looks like this: $4x + 3 = 2$. But what's the next logical step? Our mission is to get all the terms with 'x' on one side and all the constant terms (the numbers without 'x') on the other. This separation is crucial for eventually isolating 'x' and finding its value. Looking at our equation, we have a constant term '+3' on the same side as the 'x' term. To move this constant term to the other side, we need to perform an operation that will effectively cancel it out on the left side and reappear on the right side. The key here is to think about inverse operations. What's the opposite of adding 3? Subtracting 3! So, the next strategic move is to subtract 3 from both sides of the equation. This is because subtracting 3 from both sides maintains the equation's balance, a fundamental principle in algebra. When we subtract 3 from both sides, we get: $4x + 3 - 3 = 2 - 3$. On the left side, the '+3' and '-3' cancel each other out, leaving us with just $4x$. On the right side, $2 - 3$ equals $-1$. So, our equation now becomes $4x = -1$. Notice how subtracting 3 has successfully separated the variable term (4x) from the constant term. We're one step closer to solving for 'x'! By isolating the variable term, we've simplified the equation further and set ourselves up for the final step of dividing to find the value of 'x'. This strategic use of subtraction demonstrates a clear understanding of algebraic principles and how to manipulate equations to achieve our goal.

Why Subtracting 3 is the Key

Let's zoom in on why subtracting 3 is the magic move here. In the world of equations, we're always aiming for balance. Imagine a scale – whatever you add or subtract on one side, you've got to do on the other to keep it level. Our equation, $4x + 3 = 2$, is just like that scale. We want to isolate 'x', which means getting it all by itself on one side. Currently, we have that “pesky” +3 hanging out with the $4x$. It's like an extra weight on the scale, preventing us from seeing the true value of 'x'. To get rid of it, we need to perform an operation that will “cancel” it out. This is where the concept of inverse operations comes into play. The inverse operation of addition is subtraction, and vice versa. Since we have +3, we need to subtract 3 to make it disappear from the left side. But remember, we have to maintain balance! So, we subtract 3 from both sides of the equation. This is a fundamental rule in algebra – whatever you do to one side, you must do to the other. When we subtract 3 from both sides, the equation transforms. On the left, +3 and -3 cancel each other out, leaving us with just $4x$. On the right, we have $2 - 3$, which simplifies to $-1$. So, our equation becomes $4x = -1$. The +3 is gone from the left side, and we've successfully separated the variable term (4x) from the constant terms. This separation is a crucial step towards isolating 'x' and finding its value. By strategically subtracting 3, we've not only simplified the equation but also demonstrated a deep understanding of the principles of algebraic manipulation. Subtracting three helps in isolating the term with the variable, which is the main goal here. This is a fundamental strategy in solving algebraic equations.

The Final Step Isolating 'x'

We've come a long way, guys! We started with a somewhat complex equation, $-4.5x + 3 = 2 - 8.5x$, and through Quinda's clever moves and our step-by-step breakdown, we've simplified it to $4x = -1$. This is fantastic progress! But we're not quite at the finish line yet. Our ultimate goal, remember, is to find the value of 'x'. Right now, 'x' is being multiplied by 4. To get 'x' all by itself, we need to undo this multiplication. And what's the inverse operation of multiplication? Division! So, the final step in solving for 'x' is to divide both sides of the equation by 4. This is another application of the golden rule of equations: whatever you do to one side, you must do to the other to maintain balance. When we divide both sides by 4, we get: $4x / 4 = -1 / 4$. On the left side, the 4s cancel each other out, leaving us with just 'x'. On the right side, $-1 / 4$ is simply $-0.25$. So, the solution to our equation is $x = -0.25$. We've done it! We've successfully isolated 'x' and found its value. This final step of dividing highlights the importance of understanding inverse operations in algebra. By using division to undo multiplication, we were able to free 'x' and determine its value. This process demonstrates a complete mastery of the steps involved in solving linear equations.

Conclusion Mastering the Art of Equation Solving

Woo-hoo! We made it through the equation-solving journey together. From Quinda's initial move of adding $8.5x$ to both sides to our strategic subtraction and final division, we've seen how each step plays a crucial role in isolating the variable and finding the solution. Remember, solving equations is like piecing together a puzzle. Each operation you perform is a move that gets you closer to the final answer. The key is to understand the principles of balance and inverse operations. By adding, subtracting, multiplying, or dividing both sides of the equation, you can manipulate it to your advantage while keeping it balanced. And by using inverse operations, you can undo the operations that are “trapping” the variable and isolate it. So, the next time you encounter an equation, don't feel intimidated. Break it down step by step, just like we did today. Identify the operations that need to be undone, and use inverse operations to do it. Keep the equation balanced, and you'll be solving equations like a pro in no time! With practice and a clear understanding of the underlying principles, you can conquer any algebraic challenge that comes your way. Keep practicing, and you'll become a master equation solver in no time!