Solving For X: Equation With Fractions Explained

Hey guys! Today, we're diving into a mathematical problem that involves solving for x in an equation with fractions. Don't worry, it's not as intimidating as it looks! We'll break it down step by step, so you can follow along and understand exactly how to tackle these types of problems. Our main goal here is to find the value of x that makes the equation (5/2)(x + 2/3) = -5/6 true. So, grab your pencils and let's get started!

Understanding the Equation

Before we jump into the solution, let's take a closer look at the equation itself. We have (5/2)(x + 2/3) = -5/6. This might seem a bit complex at first glance, but it's actually quite manageable once we understand the different parts. We've got a fraction (5/2) multiplied by an expression in parentheses (x + 2/3), and this whole thing equals another fraction (-5/6). The key here is to use the order of operations (PEMDAS/BODMAS) and some basic algebraic principles to isolate x on one side of the equation. This means we need to undo the operations that are being done to x, one by one, until we have x all by itself. Remember, whatever we do to one side of the equation, we must also do to the other side to keep things balanced. This is the golden rule of equation solving, guys! Keeping this in mind will prevent many common errors.

The first thing we can do to simplify this equation is to distribute the 5/2 across the parentheses. This means we multiply 5/2 by both x and 2/3. By doing this, we are essentially removing the parentheses and spreading the multiplication across the terms inside. This step is crucial because it allows us to separate the x term and deal with it more directly. When we distribute, we get (5/2) * x + (5/2) * (2/3) = -5/6. Now, let's simplify these multiplications. (5/2) * x is simply (5/2)x, and (5/2) * (2/3) is (52)/(23) which simplifies to 10/6. We can further simplify 10/6 by dividing both the numerator and the denominator by their greatest common divisor, which is 2. This gives us 5/3. So, our equation now looks like (5/2)x + 5/3 = -5/6. We're making progress, guys! We've eliminated the parentheses and simplified one of the terms. The next step is to isolate the term with x, which we'll tackle in the next section.

Isolating the x Term

Now that we've simplified the equation to (5/2)x + 5/3 = -5/6, our next step is to isolate the term containing x. This means we want to get (5/2)x by itself on one side of the equation. To do this, we need to get rid of the + 5/3 that's being added to it. The way we do that is by subtracting 5/3 from both sides of the equation. Remember, whatever we do to one side, we must do to the other to maintain balance. So, we subtract 5/3 from both the left side and the right side.

This gives us (5/2)x + 5/3 - 5/3 = -5/6 - 5/3. On the left side, the + 5/3 and - 5/3 cancel each other out, leaving us with just (5/2)x. On the right side, we have -5/6 - 5/3. To subtract these fractions, we need a common denominator. The least common multiple of 6 and 3 is 6, so we'll rewrite 5/3 as an equivalent fraction with a denominator of 6. To do this, we multiply both the numerator and the denominator of 5/3 by 2, which gives us 10/6. Now we can rewrite the right side as -5/6 - 10/6. Subtracting these fractions, we get (-5 - 10)/6, which simplifies to -15/6. So, our equation now looks like (5/2)x = -15/6. We're getting closer and closer to solving for x, guys! We've isolated the x term on one side, and we've simplified the other side to a single fraction. The next step is to get rid of the coefficient (5/2) that's multiplying x.

Solving for x

We've reached the final stage! We have the equation (5/2)x = -15/6, and our goal is to find the value of x. The (5/2) is currently multiplying x, so to isolate x, we need to do the opposite operation, which is division. However, dividing by a fraction can be a bit tricky, so instead of dividing by 5/2, we'll multiply by its reciprocal. The reciprocal of a fraction is simply flipping the numerator and the denominator. So, the reciprocal of 5/2 is 2/5. We'll multiply both sides of the equation by 2/5.

This gives us (2/5) * (5/2)x = (2/5) * (-15/6). On the left side, (2/5) * (5/2) is equal to 1, so we're left with just x. On the right side, we have (2/5) * (-15/6). To multiply fractions, we multiply the numerators together and the denominators together. So, we get (2 * -15) / (5 * 6), which simplifies to -30/30. This fraction can be further simplified by dividing both the numerator and the denominator by their greatest common divisor, which is 30. This gives us -1/1, which is simply -1. Therefore, the value of x that satisfies the equation is x = -1. Woohoo! We solved it, guys! We've successfully navigated the fractions and the algebra to find the solution. It might seem like a lot of steps, but each step is logical and manageable. With practice, you'll become more comfortable with these types of problems.

Verifying the Solution

It's always a good idea to check our answer to make sure it's correct. To verify that x = -1 is indeed the solution, we substitute -1 for x in the original equation and see if it holds true. Our original equation was (5/2)(x + 2/3) = -5/6. Substituting x = -1, we get (5/2)(-1 + 2/3) = -5/6. Now, we need to simplify the expression inside the parentheses first. To add -1 and 2/3, we need a common denominator. We can rewrite -1 as -3/3. So, we have -3/3 + 2/3, which is equal to -1/3. Now our equation looks like (5/2)(-1/3) = -5/6. Multiplying the fractions on the left side, we get (5 * -1) / (2 * 3), which simplifies to -5/6. So, we have -5/6 = -5/6. This is a true statement, which confirms that our solution x = -1 is correct! High five, guys! We not only solved the equation, but we also verified our solution. This is a crucial step in problem-solving because it helps us catch any mistakes and build confidence in our answers.

Key Takeaways and Tips

So, what have we learned today? We've successfully solved for x in the equation (5/2)(x + 2/3) = -5/6. We did this by following a series of steps: distributing, isolating the x term, and using inverse operations. Here are some key takeaways and tips to keep in mind when tackling similar problems:

• Distribute: If you have parentheses, start by distributing any terms outside the parentheses to the terms inside. This simplifies the equation and makes it easier to work with.
• Isolate the x term: Get the term containing x by itself on one side of the equation. This usually involves adding or subtracting terms from both sides.
• Use inverse operations: To undo operations that are being done to x (like multiplication or addition), use the opposite operation (division or subtraction). Remember to do the same operation on both sides of the equation to maintain balance.
• Work with fractions: When dealing with fractions, make sure you have a common denominator before adding or subtracting. Multiplying fractions is straightforward: multiply the numerators and multiply the denominators.
• Simplify: Always simplify your fractions and expressions as much as possible. This makes the calculations easier and reduces the chances of making mistakes.
• Verify your solution: After you find a solution, plug it back into the original equation to make sure it works. This is a crucial step to catch any errors and ensure your answer is correct.
• Practice, practice, practice: The more you practice solving these types of equations, the more comfortable and confident you'll become. Don't be afraid to make mistakes – they're part of the learning process!

Solving equations with fractions can seem daunting at first, but with a systematic approach and a little bit of practice, you can master them. Remember to break the problem down into smaller, manageable steps, and don't be afraid to ask for help if you get stuck. Keep up the great work, guys! You're doing awesome!

Practice Problems

To solidify your understanding, here are a few practice problems you can try. Remember to follow the steps we discussed, and don't forget to verify your solutions!

1. (3/4)(x - 1/2) = 1/8
2. (2/3)(x + 5/6) = -1/3
3. (1/2)(2x - 3/4) = 5/8

Work through these problems at your own pace, and check your answers. If you have any questions or get stuck, don't hesitate to review the steps we covered or seek help from a teacher or tutor. The key is to keep practicing and building your skills. You've got this, guys!

Conclusion

Solving equations like (5/2)(x + 2/3) = -5/6 is a fundamental skill in algebra. By understanding the steps involved – distributing, isolating the x term, using inverse operations, and verifying the solution – you can confidently tackle these types of problems. Remember, guys, math is like building blocks. Each concept builds upon the previous one. So, mastering these basic skills is crucial for your future mathematical journey. Keep practicing, stay curious, and never give up. You're all capable of amazing things! Until next time, happy solving!