Solving For X: A Step-by-Step Guide To The Equation

Hey guys! Let's dive into solving a classic algebraic equation. If you've ever felt a little intimidated by fractions and variables hanging out together, don't worry! We're going to break down this equation step by step, so it feels super manageable. Our mission today is to find the value of 'x' in the equation: (1/5)x + (1/3) = (11/10)(7x - 5). Grab your pencils, and let's get started!

Understanding the Equation

Before we jump into calculations, let's make sure we understand what we're looking at. This equation is a linear equation, meaning 'x' is raised to the power of 1. These equations have at most one solution, which makes our task clear: find that single value of 'x' that makes the equation true. We have fractions involved, which might seem scary, but we'll tackle them strategically. The key is to isolate 'x' on one side of the equation. We will do this by performing the same operations on both sides, keeping everything balanced and fair. Think of it like a see-saw – whatever you do to one side, you must do to the other to maintain equilibrium. Remember those fundamental principles, and you'll be solving equations like a pro in no time!

Dealing with Fractions

The first hurdle we often encounter in equations like this is those pesky fractions. They can make things look more complicated than they are. But fear not! There's a neat trick to get rid of them: we can multiply both sides of the equation by the least common multiple (LCM) of the denominators. In our equation, the denominators are 5, 3, and 10. So, what's the LCM of 5, 3, and 10? Well, let's break it down. The multiples of 5 are 5, 10, 15, 20, 25, 30… The multiples of 3 are 3, 6, 9, 12, 15, 18, 21, 24, 27, 30… And the multiples of 10 are 10, 20, 30… The smallest number that appears in all three lists is 30. So, the LCM is 30! Now, we're going to multiply every term in the equation by 30. This is crucial: we need to distribute the multiplication across all terms to maintain the equation's balance. This step is going to clear out those fractions and make our equation much easier to handle. Trust me, this is a game-changer!

Multiplying Through by the LCM

Okay, we've identified our LCM as 30. Now comes the fun part – multiplying! We'll multiply both sides of the equation by 30. On the left side, we have (1/5)x + (1/3). So, we'll multiply both (1/5)x and (1/3) by 30. On the right side, we have (11/10)(7x - 5), so we'll multiply that entire expression by 30. Let's break it down piece by piece. 30 * (1/5)x is the same as (30/5)x, which simplifies to 6x. Next, 30 * (1/3) is the same as (30/3), which simplifies to 10. So, the left side of our equation now looks like 6x + 10. Now, for the right side, we have 30 * (11/10)(7x - 5). We can think of this as (30/10) * 11 * (7x - 5), which simplifies to 3 * 11 * (7x - 5), or 33(7x - 5). See how those fractions are disappearing? It's like magic! We've successfully transformed our equation into one without fractions, making it much friendlier to solve. Remember to take your time with these multiplications and double-check your work to avoid any silly mistakes. Accuracy is key in algebra!

Simplifying the Equation

Great! We've cleared out the fractions, and our equation is looking much cleaner. Now, it's time to simplify further. Our equation currently looks like this: 6x + 10 = 33(7x - 5). The next step is to tackle the parentheses on the right side. We'll use the distributive property, which means we multiply the 33 by both the 7x and the -5 inside the parentheses. So, 33 * 7x is 231x, and 33 * -5 is -165. That means our equation now reads: 6x + 10 = 231x - 165. We're making progress! We've expanded the equation, and now we have 'x' terms and constant terms on both sides. The goal is still to isolate 'x', but to do that, we need to gather all the 'x' terms on one side and all the constant terms on the other. This is like sorting socks – we want all the matching pairs together. We'll use addition and subtraction to move terms across the equals sign, remembering to do the same operation on both sides to keep everything balanced.

Isolating the Variable

Alright, let's get those 'x' terms together! We have 6x on the left side and 231x on the right side. It's usually easier to move the smaller 'x' term to the side with the larger 'x' term to avoid dealing with negative coefficients. So, we'll subtract 6x from both sides of the equation. This gives us: 6x + 10 - 6x = 231x - 165 - 6x, which simplifies to 10 = 225x - 165. Now, let's gather the constant terms. We have 10 on the left side and -165 on the right side. To move the -165 to the left side, we'll add 165 to both sides of the equation. This gives us: 10 + 165 = 225x - 165 + 165, which simplifies to 175 = 225x. We're getting so close! We have all the 'x' terms on one side and all the constant terms on the other. Now, we just need to get 'x' by itself. Currently, 'x' is being multiplied by 225. To undo this multiplication, we'll divide both sides of the equation by 225. Remember, whatever we do to one side, we must do to the other!

Finding the Value of x

We're in the home stretch now! Our equation looks like this: 175 = 225x. To isolate 'x', we need to divide both sides by 225. So, we have 175 / 225 = (225x) / 225, which simplifies to x = 175 / 225. Now, let's simplify that fraction. Both 175 and 225 are divisible by 25. 175 divided by 25 is 7, and 225 divided by 25 is 9. So, our fraction simplifies to x = 7/9. We did it! We've found the value of 'x' that satisfies the equation. It might have seemed like a long journey, but we tackled it step by step, and we arrived at the solution. Remember, the key to solving equations is to stay organized, perform the same operations on both sides, and break the problem down into smaller, manageable steps. You've got this!

Checking Our Solution

It's always a good idea to check our solution to make sure we didn't make any mistakes along the way. To do this, we'll substitute x = 7/9 back into the original equation: (1/5)x + (1/3) = (11/10)(7x - 5). Let's plug in 7/9 for 'x'. On the left side, we have (1/5)(7/9) + (1/3), which is 7/45 + 1/3. To add these fractions, we need a common denominator, which is 45. So, we rewrite 1/3 as 15/45. Now we have 7/45 + 15/45, which equals 22/45. On the right side, we have (11/10)(7(7/9) - 5). First, let's simplify inside the parentheses. 7 * (7/9) is 49/9. So, we have 49/9 - 5. To subtract these, we need a common denominator, which is 9. We rewrite 5 as 45/9. Now we have 49/9 - 45/9, which equals 4/9. So, the right side becomes (11/10)(4/9), which is 44/90. We can simplify this fraction by dividing both the numerator and denominator by 2, giving us 22/45. Guess what? The left side (22/45) equals the right side (22/45)! That means our solution, x = 7/9, is correct. Woohoo! You've successfully solved a linear equation with fractions and verified your answer. Keep practicing, and you'll become an equation-solving master!

Conclusion

So, there you have it, guys! We've successfully navigated the equation (1/5)x + (1/3) = (11/10)(7x - 5) and found that x = 7/9. We tackled fractions, distributed terms, isolated the variable, and even checked our solution to be extra sure. Remember, solving equations is like building a puzzle – each step brings you closer to the final picture. The key is to break down the problem, stay organized, and keep practicing. Don't be afraid of fractions or parentheses; they're just part of the adventure! Keep honing your skills, and you'll be solving even more complex equations with confidence. You've got this! Keep up the awesome work, and I'll catch you in the next math challenge!