Solving For X: A Step-by-Step Guide

Hey math enthusiasts! Today, we're diving into a classic algebra problem: solving for x in the equation $\displaystyle x+\frac{\frac{5x-1}{3} -2}{2} =\frac{x}{5} +\frac{7}{15}$. Don't worry if it looks a bit intimidating at first – we'll break it down into easy-to-follow steps. Getting a good grasp of this type of problem is super important because it's a fundamental concept in algebra and shows up everywhere in higher math. So, let's get started and unravel this equation together! We'll use a clear, concise method to find the value of x.

Step 1: Simplify the Equation

Our first order of business is to simplify the equation. This means getting rid of fractions within fractions and making things a little less cluttered. Look at the equation again: $\displaystyle x+\frac{\frac{5x-1}{3} -2}{2} =\frac{x}{5} +\frac{7}{15}$. We'll start by simplifying the complex fraction on the left side. The goal here is to make the equation easier to work with, which means fewer fractions and simpler terms. Let's tackle that fraction step by step, focusing on the left side of the equation first.

First, deal with the numerator of the inner fraction: $\frac{5x-1}{3} -2$. To subtract 2, we need a common denominator. Since 2 is the same as $\frac{6}{3}$, we can rewrite this as $\frac{5x-1}{3} - \frac{6}{3}$. Now, subtract the numerators: $\frac{5x-1-6}{3}$, which simplifies to $\frac{5x-7}{3}$. This simplifies the left side a bit more, making the problem easier to solve. Now, the left side of the original equation looks like this: $\frac{\frac{5x-7}{3}}{2}$. This is the same as $(\frac{5x-7}{3}) / 2$. To divide by 2, we multiply by $\frac{1}{2}$, resulting in $\frac{5x-7}{6}$. Thus, our equation becomes $\displaystyle x+\frac{5x-7}{6} =\frac{x}{5} +\frac{7}{15}$. We've simplified a lot! We've transformed the complex fraction into a much more manageable expression. Good job so far, guys!

Step 2: Eliminate Fractions

Okay, so we've got our simplified equation: $\displaystyle x+\frac{5x-7}{6} =\frac{x}{5} +\frac{7}{15}$. Next up, we need to eliminate those pesky fractions. The easiest way to do this is to multiply every term in the equation by the least common multiple (LCM) of the denominators. In this case, the denominators are 6, 5, and 15. The LCM of 6, 5, and 15 is 30. Get your pencils ready, because we're about to multiply each term by 30. This means taking each term and multiplying it by 30, making sure to apply the distributive property correctly. This is a crucial step to avoid mistakes.

Let's go through it step by step. First, multiply x by 30: 30x = 30x. Next, multiply $\frac{5x-7}{6}$ by 30: 30 * $\frac{5x-7}{6}$ = 5 * (5x - 7) = 25x - 35. Then, multiply $\frac{x}{5}$ by 30: 30 * $\frac{x}{5}$ = 6x. Finally, multiply $\frac{7}{15}$ by 30: 30 * $\frac{7}{15}$ = 2 * 7 = 14. Now, put it all together. Your equation becomes: 30x + 25x - 35 = 6x + 14. See how much cleaner the equation looks without those fractions? It's much easier to work with now. By multiplying by the LCM, we've transformed the equation into one that's much simpler to solve, allowing us to focus on isolating x.

Step 3: Combine Like Terms

Alright, now that we've banished the fractions, let's combine like terms to tidy things up even further. Remember the equation from the last step: 30x + 25x - 35 = 6x + 14. Combining like terms means adding or subtracting terms that have the same variable (in this case, x) and combining constant terms (the numbers without variables). This process simplifies both sides of the equation, making it easier to isolate x. Let's start with the left side of the equation. We have two terms with x: 30x and 25x. Add them together: 30x + 25x = 55x. The left side now looks like this: 55x - 35. The left side is simplified, now let's move on to the right side of the equation. There's only one term with x: 6x, and we also have a constant term of 14. The right side remains as is: 6x + 14. So, the whole equation now is: 55x - 35 = 6x + 14. This is a lot easier to work with. We've simplified both sides of the equation by combining like terms. This sets us up for the next step, where we'll isolate x even further.

Step 4: Isolate the Variable

Okay, it's time to isolate the variable x. This means getting all the terms with x on one side of the equation and all the constant terms on the other side. Think of it like a game of tug-of-war where we want to get x all by itself. We currently have the equation: 55x - 35 = 6x + 14. Let's start by getting all the x terms on the left side. To do this, we need to subtract 6x from both sides of the equation. When you subtract 6x from the right side, it cancels out the 6x. When you subtract 6x from the left side, you get: 55x - 6x - 35 = 14. Simplify the left side to get: 49x - 35 = 14. Great job! The equation is getting closer to our final answer. Now we have 49x - 35 = 14. To continue isolating x, we need to get rid of that -35 on the left side. We do this by adding 35 to both sides of the equation. This isolates the variable even further and brings us closer to finding the value of x.

Adding 35 to both sides: 49x - 35 + 35 = 14 + 35. Simplifying this, we get 49x = 49. Now we are close to the finish line. Our goal is to get x alone on one side of the equation and a number on the other. It is much easier to work with now. Adding 35 to both sides gives us a simpler equation to solve.

Step 5: Solve for x

We're in the final stretch now! We've got our simplified equation: 49x = 49. The last step is to solve for x. To get x all by itself, we need to divide both sides of the equation by the coefficient of x, which is 49 in this case. Dividing both sides by 49, we get: (49x) / 49 = 49 / 49. When you divide 49x by 49, you're left with just x on the left side. When you divide 49 by 49, you get 1. So, the equation becomes x = 1. Congratulations, guys! We've successfully solved for x.

Step 6: The Answer

Therefore, the value of x is 1. We went through each step carefully, from simplifying the original equation to isolating x and finally solving for its value. The answer is x = 1. We can always double-check our work by plugging x = 1 back into the original equation to ensure it's correct.

In Summary

1. Simplify: Handle nested fractions.
2. Eliminate Fractions: Multiply by the least common multiple (LCM).
3. Combine Like Terms: Group x terms and constant terms.
4. Isolate x: Move x terms to one side and constants to the other.
5. Solve for x: Divide by the coefficient of x.

Keep practicing, and you'll become a pro at these types of problems in no time. If you got stuck at any point, go back and review that step again. Good luck and happy solving!