Solving X/4 – 7/5 + X/10 = X/2 + 7/20 A Step-by-Step Guide

by Omar Yusuf 59 views

Hey guys! Today, we're diving deep into solving a classic algebraic equation. We've got a poser here: X/4 – 7/5 + X/10 = X/2 + 7/20. Don't worry if it looks intimidating at first glance; we're going to break it down step by step, so it's super easy to follow. Our main goal here is to isolate 'X' on one side of the equation. To do this effectively, we'll need to get rid of those pesky fractions. Think of it as clearing a path so we can see 'X' clearly! Let's jump right in and turn this equation from a monster into a manageable math problem. Remember, the key to algebra is patience and persistence. We'll tackle each part methodically, and by the end of this guide, you'll be a pro at solving equations like this one. So, grab your pencils, notebooks, and let's get started! We're going to transform this equation together, making sure every step is crystal clear. Let's not just find the answer; let's understand the process. That's the real magic of math – knowing why we do what we do. So, are you ready to unravel this equation and make 'X' our own? Let's do it!

Step 1: Finding the Least Common Denominator (LCD)

Okay, team, before we can really start moving things around, we need to tackle those fractions. The best way to do this is by finding the Least Common Denominator, or LCD. Think of the LCD as the magic number that will allow us to add and subtract fractions easily. In our equation, X/4 – 7/5 + X/10 = X/2 + 7/20, we have the denominators 4, 5, 10, and 20. So, how do we find the LCD? Well, we need to find the smallest number that each of these denominators can divide into evenly. Let's list out the multiples of each number:

  • Multiples of 4: 4, 8, 12, 16, 20, 24...
  • Multiples of 5: 5, 10, 15, 20, 25...
  • Multiples of 10: 10, 20, 30, 40...
  • Multiples of 20: 20, 40, 60, 80...

See that? The smallest number that appears in all of these lists is 20. So, our LCD is 20! Now, why is this so important? Because we can multiply both sides of the equation by 20, which will eliminate the fractions. This is like giving our equation a super-clean makeover! It'll make the next steps much easier to handle. Imagine trying to cook with a messy kitchen versus a clean one. Same idea here! Finding the LCD is like organizing our ingredients before we start cooking up the solution. It sets us up for success and makes the whole process smoother. So, now that we have our LCD, 20, we're ready to move on to the next step and start clearing those fractions. Let's keep the momentum going!

Step 2: Multiplying Both Sides of the Equation by the LCD

Alright, now that we've got our LCD, which is 20, the real fun begins! This is where we start to see those fractions disappear. We're going to multiply both sides of the equation X/4 – 7/5 + X/10 = X/2 + 7/20 by 20. Remember, whatever we do to one side of the equation, we absolutely have to do to the other side. It's like a balancing act – we need to keep things fair and square! So, let's break it down:

20 * (X/4 – 7/5 + X/10) = 20 * (X/2 + 7/20)

Now, we'll distribute the 20 to each term inside the parentheses:

(20 * X/4) – (20 * 7/5) + (20 * X/10) = (20 * X/2) + (20 * 7/20)

See how we're multiplying 20 by each fraction? This is where the magic happens! Now, let's simplify each term:

  • (20 * X/4) = 5X (Because 20 divided by 4 is 5)
  • (20 * 7/5) = 28 (Because 20 divided by 5 is 4, and 4 times 7 is 28)
  • (20 * X/10) = 2X (Because 20 divided by 10 is 2)
  • (20 * X/2) = 10X (Because 20 divided by 2 is 10)
  • (20 * 7/20) = 7 (Because 20 divided by 20 is 1)

So, our equation now looks like this: 5X – 28 + 2X = 10X + 7. How much cleaner is that? We've gotten rid of all the fractions, and we're left with a much simpler equation to work with. This is a huge step forward! Multiplying by the LCD is like using a powerful cleaning tool to wipe away all the clutter. Now we can see the equation in its simplest form, which makes the next steps much easier. We're on a roll! Let's move on to the next step: combining like terms.

Step 3: Combining Like Terms

Okay, we've successfully cleared the fractions, and our equation now looks like this: 5X – 28 + 2X = 10X + 7. The next step is to gather all the similar elements – it's like organizing your closet by putting shirts with shirts and pants with pants. We call this combining like terms. On the left side of the equation, we have two terms with 'X': 5X and 2X. We can add these together. Think of it as having 5 apples and then getting 2 more apples – you now have 7 apples! So, 5X + 2X = 7X. Now our equation looks like this:

7X – 28 = 10X + 7

See how we've simplified the left side? Now, let's look at the right side of the equation. We have 10X + 7. There are no other 'X' terms or constant terms to combine on this side, so we'll leave it as it is. Combining like terms is super important because it makes the equation much easier to handle. It's like taking a big, messy problem and breaking it down into smaller, more manageable pieces. By combining like terms, we're making our equation more streamlined and easier to solve. We've essentially tidied up the equation, making it much clearer what our next steps should be. So, now that we've combined like terms, our equation is in a great shape. We're ready to move on to the next step, which is getting all the 'X' terms on one side and the constants on the other. Let's keep up the great work!

Step 4: Moving Variables to One Side

Alright, we're making fantastic progress! Our equation is currently 7X – 28 = 10X + 7. Now, we need to get all the terms with 'X' on one side of the equation and all the constant terms (the numbers without 'X') on the other side. Think of it like sorting your socks – you want all the matching pairs together. It doesn't matter which side we choose for the 'X' terms, but it's often easier to keep the coefficient (the number in front of 'X') positive if we can. In this case, we have 7X on the left and 10X on the right. To keep the coefficient positive, let's move the 7X to the right side. To do this, we'll subtract 7X from both sides of the equation. Remember, whatever we do to one side, we have to do to the other to keep things balanced! So:

7X – 28 – 7X = 10X + 7 – 7X

On the left side, 7X – 7X cancels out, leaving us with -28. On the right side, 10X – 7X simplifies to 3X. So, our equation now looks like this:

-28 = 3X + 7

See how we've successfully moved the 'X' term to the right side? We're one step closer to isolating 'X'! Moving variables to one side is a crucial step in solving equations. It's like organizing your ingredients before you start cooking – you need everything in its place to make the process smoother. By getting all the 'X' terms on one side, we're simplifying the equation and making it easier to see what we need to do next. We're doing great! Now, let's move on to the next step and isolate the 'X' even further by dealing with the constant term on the right side.

Step 5: Isolating the Variable

We're on the home stretch now! Our equation is currently -28 = 3X + 7. Our goal is to get 'X' all by itself on one side of the equation. Think of it like giving 'X' its own spotlight – we want it to be the star of the show! To do this, we need to get rid of the +7 that's hanging out with the 3X on the right side. We can do this by subtracting 7 from both sides of the equation. Again, remember the balance – whatever we do to one side, we must do to the other:

-28 – 7 = 3X + 7 – 7

On the left side, -28 – 7 equals -35. On the right side, +7 and -7 cancel each other out, leaving us with just 3X. So, our equation now looks like this:

-35 = 3X

We're so close! We've managed to isolate the term with 'X' on the right side. Now, there's just one more step to get 'X' completely alone. We have -35 = 3X, which means 3 times 'X' equals -35. To find out what 'X' is, we need to undo this multiplication. How do we undo multiplication? We divide! We'll divide both sides of the equation by 3:

-35 / 3 = 3X / 3

On the right side, 3X divided by 3 is simply 'X'. On the left side, -35 divided by 3 is -35/3, which is a fraction that doesn't simplify nicely. So, our solution is:

X = -35/3

We did it! We've successfully isolated 'X' and found its value. Give yourself a pat on the back – you've conquered this equation! Isolating the variable is the heart of solving algebraic equations. It's like the final puzzle piece falling into place. By carefully undoing the operations that are attached to 'X', we reveal its true value. We've taken a complex equation and broken it down into manageable steps, and now we have our answer. But we're not quite done yet – there's one more crucial step to make sure we've truly solved the equation.

Step 6: Checking the Solution

Okay, we've found our solution: X = -35/3. But before we declare victory, it's super important to make sure our answer is correct. Think of it like proofreading your work before you hand it in – you want to catch any mistakes! To check our solution, we'll substitute -35/3 back into the original equation: X/4 – 7/5 + X/10 = X/2 + 7/20. So, let's plug in -35/3 for X:

(-35/3) / 4 – 7/5 + (-35/3) / 10 = (-35/3) / 2 + 7/20

This looks a bit messy, but don't worry, we'll take it step by step. Remember that dividing by a number is the same as multiplying by its reciprocal. So, (-35/3) / 4 is the same as (-35/3) * (1/4), which equals -35/12. Similarly, (-35/3) / 10 is the same as (-35/3) * (1/10), which equals -35/30, which simplifies to -7/6. And (-35/3) / 2 is the same as (-35/3) * (1/2), which equals -35/6. Now our equation looks like this:

-35/12 – 7/5 – 7/6 = -35/6 + 7/20

Ugh, still lots of fractions! To make things easier, let's find the LCD of all these fractions. The denominators are 12, 5, 6, and 20. The LCD is 60. So, we'll convert each fraction to have a denominator of 60:

  • -35/12 = -175/60
  • -7/5 = -84/60
  • -7/6 = -70/60
  • -35/6 = -350/60
  • 7/20 = 21/60

Now our equation looks like this:

-175/60 – 84/60 – 70/60 = -350/60 + 21/60

Let's add the fractions on each side:

  • Left side: -175/60 – 84/60 – 70/60 = -329/60
  • Right side: -350/60 + 21/60 = -329/60

Guess what? Both sides are equal! -329/60 = -329/60. This means our solution, X = -35/3, is correct! Woohoo! Checking our solution is the ultimate way to ensure we've done everything right. It's like the final stamp of approval on our work. By substituting our solution back into the original equation and verifying that both sides are equal, we can be confident that we've solved the equation accurately. And that's a fantastic feeling!

Conclusion

Guys, we did it! We successfully solved the equation X/4 – 7/5 + X/10 = X/2 + 7/20. We found the LCD, cleared the fractions, combined like terms, isolated the variable, and even checked our solution. You've shown some serious math skills today! Solving algebraic equations can seem daunting at first, but by breaking them down into smaller, manageable steps, we can tackle even the trickiest problems. Remember, the key is to stay organized, be patient, and double-check your work. Math isn't just about getting the right answer; it's about understanding the process and building problem-solving skills. So, keep practicing, keep exploring, and never be afraid to ask questions. You're all math superstars in the making! And remember, every equation you solve makes you a little bit stronger and more confident. So, go out there and conquer those math challenges! You've got this!