Solving 6x - 2 = -4x + 2: Spencer Vs. Jeremiah
Hey guys! Let's dive into a common algebra problem where we need to figure out the best way to solve an equation. Today, we’re looking at the equation 6x - 2 = -4x + 2. We've got two students, Spencer and Jeremiah, each with their own idea on how to start solving it. Spencer thinks we should add 4x to both sides, while Jeremiah believes subtracting 6x from both sides is the way to go. So, who's right? Or are they both on the right track? Let’s break it down and see what’s happening step by step.
Understanding the Initial Steps
When you first look at the equation 6x - 2 = -4x + 2, it might seem a bit daunting, but don't worry, we'll make it super clear. The main goal when solving any algebraic equation is to isolate the variable—in this case, x. This means we want to get x all by itself on one side of the equals sign. To do this, we need to undo any operations that are being done to x, like addition, subtraction, multiplication, or division. The golden rule here is that whatever you do to one side of the equation, you must do to the other side to keep everything balanced. Think of it like a scale; if you add weight to one side, you need to add the same amount to the other to keep it level. So, both Spencer and Jeremiah are thinking about how to move those x terms around, but they’ve chosen different operations to get started. The beauty of algebra is that there isn't always just one 'right' way to start; sometimes, multiple paths can lead to the same correct answer. Let's explore each of their approaches and see why they both make sense.
Spencer's Approach: Adding 4x to Both Sides
Spencer's idea is to add 4x to both sides of the equation 6x - 2 = -4x + 2. This is a solid move, and here’s why. By adding 4x to both sides, Spencer is aiming to eliminate the x term on the right side of the equation. Remember, we want to isolate x, and getting rid of the -4x on the right side is a step in that direction. When we add 4x to -4x, they cancel each other out, leaving us with just the constant term on that side. So, let's see it in action:
Starting with our equation:
6x - 2 = -4x + 2
Now, add 4x to both sides:
(6x - 2) + 4x = (-4x + 2) + 4x
Combine like terms. On the left side, we combine 6x and 4x to get 10x. On the right side, -4x and 4x cancel each other out:
10x - 2 = 2
See how the equation is already looking simpler? We've managed to get all the x terms on one side, which is excellent progress. Now, we're just a couple of steps away from fully isolating x. Spencer’s approach is totally valid and a smart way to kick things off. It helps consolidate the x terms, making the equation easier to manage. There's no algebraic foul play here; just a good, clean move toward solving for x!
Jeremiah's Approach: Subtracting 6x from Both Sides
Now let's consider Jeremiah's suggestion: subtracting 6x from both sides of the equation 6x - 2 = -4x + 2. This might seem like a different route than Spencer’s, but it’s equally valid. Jeremiah’s strategy here is to eliminate the x term on the left side of the equation. By subtracting 6x from both sides, he's trying to move all the x terms to the right side. There’s no right or wrong choice at this stage; it’s just about preference and what makes the most sense to you. Let’s see how this plays out step by step:
Starting with our equation:
6x - 2 = -4x + 2
Subtract 6x from both sides:
(6x - 2) - 6x = (-4x + 2) - 6x
Combine like terms. On the left side, 6x and -6x cancel each other out, leaving us with just -2. On the right side, we combine -4x and -6x to get -10x:
-2 = -10x + 2
Again, we've made progress. We’ve shifted the x terms to the right side, and now we have an equation that’s closer to being solved. Jeremiah’s approach is just as legitimate as Spencer’s. It’s all about using those algebraic moves to rearrange the equation in a way that helps you isolate the variable. There's no algebraic rule that says you must eliminate x from one side before the other. Jeremiah’s method is perfectly sound and demonstrates a good understanding of equation manipulation.
Are Both Approaches Correct?
So, let's get to the heart of the matter: Are both Spencer and Jeremiah correct in their initial steps? The answer is a resounding yes! Both approaches are perfectly valid ways to start solving the equation 6x - 2 = -4x + 2. This is one of the cool things about algebra – often, there isn't just one single path to the solution. What matters is that you follow the rules of algebra (like doing the same operation on both sides) and work towards isolating the variable. Spencer chose to add 4x to both sides, which eliminates the -4x term on the right. Jeremiah chose to subtract 6x from both sides, which eliminates the 6x term on the left. Both of these moves are algebraically sound and bring us closer to solving for x. The key takeaway here is that different approaches can be equally correct, and it’s all about finding the method that clicks best for you. Whether you prefer to move the x terms to the left or the right, as long as you're consistent and accurate with your operations, you'll arrive at the correct solution. It’s like choosing a route to a destination – multiple roads can get you there!
Continuing to Solve the Equation: Spencer's Route
Let's follow through with Spencer's approach to see how the rest of the solution unfolds. Remember, after Spencer added 4x to both sides of the original equation 6x - 2 = -4x + 2, we arrived at:
10x - 2 = 2
Now, we need to continue isolating x. The next step is to get rid of the -2 on the left side. We can do this by adding 2 to both sides of the equation:
(10x - 2) + 2 = 2 + 2
This simplifies to:
10x = 4
We're almost there! Now, we have 10x equals 4. To get x by itself, we need to undo the multiplication. We do this by dividing both sides by 10:
10x / 10 = 4 / 10
This gives us:
x = 4/10
We can simplify the fraction 4/10 by dividing both the numerator and the denominator by their greatest common divisor, which is 2:
x = 2/5
So, using Spencer’s starting move, we found that x = 2/5. Pretty neat, right? Each step we took was about reversing the operations to peel away the layers around x until it stood alone. By following the rules of algebra and staying consistent, we arrived at our solution. It’s like a mathematical puzzle, and each step is a piece falling into place!
Continuing to Solve the Equation: Jeremiah's Route
Now, let's see how the solution progresses starting with Jeremiah's approach. After Jeremiah subtracted 6x from both sides of 6x - 2 = -4x + 2, we had:
-2 = -10x + 2
Our goal remains the same: to isolate x. First, we need to get rid of the +2 on the right side of the equation. We can do this by subtracting 2 from both sides:
-2 - 2 = (-10x + 2) - 2
This simplifies to:
-4 = -10x
Now, we have -4 equals -10x. To isolate x, we need to undo the multiplication by -10. We do this by dividing both sides by -10:
-4 / -10 = -10x / -10
This gives us:
x = 4/10
Just like with Spencer’s approach, we can simplify the fraction 4/10 by dividing both the numerator and the denominator by 2:
x = 2/5
Guess what? We arrived at the same solution: x = 2/5! This perfectly illustrates that different starting points, when handled correctly, lead to the same destination. Jeremiah’s path was a bit different, but by consistently applying algebraic principles, we successfully isolated x and found its value. It's like taking a scenic route – you might see different sights along the way, but you still end up at the same place. This is why understanding the underlying rules is so crucial in math; it gives you the flexibility to solve problems in a way that makes sense to you.
Conclusion
In summary, both Spencer and Jeremiah are correct in their initial steps to solve the equation 6x - 2 = -4x + 2. Spencer's approach of adding 4x to both sides and Jeremiah's approach of subtracting 6x from both sides are both valid and lead to the same solution. The beauty of algebra is that there are often multiple paths to the correct answer. What’s most important is understanding the underlying principles and applying them consistently. Whether you choose to eliminate the x term from the left or the right side first, the key is to maintain balance by performing the same operations on both sides of the equation. By doing so, you’ll successfully isolate the variable and find the solution. In this case, we found that x = 2/5, regardless of which method we started with. So, next time you're faced with an equation, remember that you have options and that the journey to the solution can be just as insightful as the answer itself! Keep exploring, keep practicing, and you'll become a master of algebra in no time!
