Identifying Equations With The Same Solution Set As (2/3) - X + (1/6) = 6x
Hey guys! Today, we're diving into the fascinating world of equations and their solutions. Specifically, we're going to explore how to identify equations that share the same solution set without actually solving them. That's right, we're going to use our understanding of mathematical properties to navigate through equations like pros. Our main focus is the equation (2/3) - x + (1/6) = 6x, and we'll be determining which of the given equations have the same solution as this one. This is a super useful skill because it allows us to simplify problems and save time. So, let's jump in and unlock the secrets of equation equivalence!
Let's start by taking a closer look at our original equation: (2/3) - x + (1/6) = 6x. This equation is a linear equation, meaning it involves variables raised to the power of 1. To figure out which other equations have the same solution set, we need to understand how we can manipulate this equation without changing its fundamental solution. Think of it like this: we're looking for equations that are essentially the same equation in disguise. We can add, subtract, multiply, or divide both sides of the equation by the same non-zero value, and the solution will remain the same. This is based on the fundamental properties of equality. Our goal here is not to solve for x directly, but rather to transform the original equation into different forms and see if those forms match any of the given options. This approach emphasizes conceptual understanding over rote calculation, which is crucial for developing strong mathematical skills. Remember, math isn't just about finding the answer; it's about understanding why the answer is what it is.
Simplifying the Original Equation
Before we start comparing, let's simplify the original equation a bit. We have fractions involved, which can sometimes make things look more complicated than they are. Let's get rid of those fractions! The denominators in our equation are 3 and 6. The least common multiple of 3 and 6 is 6. So, we can multiply both sides of the equation by 6. This will clear the fractions and give us a cleaner equation to work with. Multiplying both sides of (2/3) - x + (1/6) = 6x by 6, we get:
6 * [(2/3) - x + (1/6)] = 6 * (6x)
Distributing the 6 on the left side, we have:
6 * (2/3) - 6 * x + 6 * (1/6) = 36x
This simplifies to:
4 - 6x + 1 = 36x
Now our equation looks much simpler! This is a key step in our process because this simplified form will be easier to compare with the other equations. Always remember, simplifying equations is a powerful tool in your mathematical arsenal. It not only makes equations easier to solve but also helps in identifying equivalent forms. So, we've transformed our original equation into 4 - 6x + 1 = 36x. Keep this in mind as we move on to analyze the other options.
Okay, now that we've simplified our original equation, 4 - 6x + 1 = 36x, let's put on our detective hats and examine the other equations. We're looking for equations that can be manipulated to look exactly like our simplified original equation. Remember, we can use the properties of equality to add, subtract, multiply, or divide both sides of an equation by the same thing without changing its solution set. This is our superpower in this quest! We'll go through each equation one by one, comparing it to our simplified original equation and seeing if they match up. Let's get started!
Equation 1: 4 - 6x + 1 = 36x
The first equation on our list is 4 - 6x + 1 = 36x. Wait a minute... This looks awfully familiar, doesn't it? In fact, it's exactly the same as the simplified form of our original equation! We derived this equation by multiplying both sides of the original equation by 6. Since it's identical, we can confidently say that this equation has the same solution set as our original equation. This is a great example of how simplifying an equation can reveal its relationship to other equations. Sometimes, the answer is staring you right in the face, but you need to do a little bit of transformation to see it clearly. So, we've got our first match! Let's move on to the next equation and see if it's another hidden twin.
Equation 2: (5/6) - x = 6x
Next up, we have the equation (5/6) - x = 6x. This equation looks a bit different from our simplified original equation, 4 - 6x + 1 = 36x. But don't let that fool you! We need to see if we can manipulate this equation to look like our original. The key here is to remember what we did in the first simplification step: multiplying by a common denominator to clear fractions. If we can get this equation to look like our original after multiplying by 6, then we know they have the same solution set. So, let's multiply both sides of (5/6) - x = 6x by 6:
6 * [(5/6) - x] = 6 * (6x)
Distributing the 6 on the left side, we get:
6 * (5/6) - 6 * x = 36x
Simplifying, we have:
5 - 6x = 36x
Now, let's compare this to our original simplified equation, 4 - 6x + 1 = 36x. Notice something interesting? The right sides of both equations are the same (36x), and the left sides have a similar term (-6x). The only difference is the constant term. On our simplified original equation, we have 4 + 1 = 5, and in this equation, we directly have 5. So, these equations are indeed equivalent! This equation also shares the same solution set as our original equation. We're on a roll!
Equation 3: 4 - x + 1 = 6x
Let's tackle the third equation: 4 - x + 1 = 6x. Comparing this to our simplified original equation, 4 - 6x + 1 = 36x, we can see a clear difference. The left side has similar constant terms (4 and 1), but the x term is different. In our original equation, we have -6x, but here we have -x. Also, the right side is 6x in this equation, while it's 36x in our original. It seems unlikely that we can manipulate this equation to match our original. To be sure, we can try to rearrange the terms, but it's clear that the coefficients of x are different, and there's no simple way to transform -x into -6x while simultaneously transforming 6x into 36x. So, this equation does not have the same solution set as our original equation. It's important to recognize these differences early on to avoid wasting time on unnecessary manipulations.
Equation 4: (5/6) + x = 6x
Now, let's consider the fourth equation: (5/6) + x = 6x. Again, we need to compare this to our original simplified equation, 4 - 6x + 1 = 36x. The presence of a fraction (5/6) suggests that we might want to multiply both sides by 6, similar to what we did before. So, let's try that:
6 * [(5/6) + x] = 6 * (6x)
Distributing the 6 on the left side gives us:
6 * (5/6) + 6 * x = 36x
Simplifying, we get:
5 + 6x = 36x
Comparing this to our simplified original equation, 4 - 6x + 1 = 36x (which simplifies to 5 - 6x = 36x), we see a significant difference. In this equation, we have +6x on the left side, while in our original, we have -6x. This sign difference is crucial. There's no way to change +6x into -6x by simply adding, subtracting, multiplying, or dividing both sides of the equation by a constant. Therefore, this equation does not share the same solution set as our original equation. It's essential to pay close attention to signs when comparing equations, as they can drastically change the solution set.
Equation 5: 5 = 30x
Let's examine the fifth equation: 5 = 30x. To determine if this equation has the same solution set as our original equation, 4 - 6x + 1 = 36x (which simplifies to 5 - 6x = 36x), we need to see if we can manipulate it to look like our original. Notice that this equation is quite simple. We have a constant on one side and a term with x on the other. To make a fair comparison, let's isolate the constant term in our simplified original equation. We can do this by adding 6x to both sides:
5 - 6x + 6x = 36x + 6x
This simplifies to:
5 = 42x
Now, let's compare 5 = 30x with 5 = 42x. The left sides are the same, but the coefficients of x on the right side are different (30 versus 42). This tells us that the solutions for x in these two equations will be different. Therefore, the equation 5 = 30x does not have the same solution set as our original equation.
Equation 6: 5 = 42x
Finally, let's look at the last equation: 5 = 42x. We actually just derived this equation while analyzing the previous equation! We took our simplified original equation, 4 - 6x + 1 = 36x (which simplifies to 5 - 6x = 36x), and added 6x to both sides to get 5 = 42x. Since we obtained this equation directly from our original equation through valid algebraic manipulations, it must have the same solution set. This is a great example of how working through the problem step-by-step can lead you to the answer. We didn't even need to do any extra work here because we had already done the transformation in the previous step!
Alright, guys, we've reached the end of our equation exploration! We started with the equation (2/3) - x + (1/6) = 6x and set out to find which of the given equations have the same solution set. By using our knowledge of algebraic properties and equation manipulation, we were able to identify the equations that are secretly the same as our original. We found that the following equations share the same solution set:
- 4 - 6x + 1 = 36x
- (5/6) - x = 6x
- 5 = 42x
This exercise highlights the power of understanding mathematical properties. Instead of solving each equation individually (which would have been much more time-consuming), we used our understanding of equivalence to identify the matches. This is a valuable skill that will serve you well in more advanced math courses and in problem-solving in general. Remember, math is not just about finding the right answer; it's about understanding the why behind the answer. Keep practicing, keep exploring, and you'll become equation masters in no time!