Solving Equations: A Step-by-Step Guide

Hey everyone! Today, we're diving into the world of algebra to solve a classic equation: $-\frac{1}{2}\left(3 x-\frac{1}{3}\right)+x=4$. Don't worry, it might look a little intimidating at first, but trust me, we'll break it down step by step to make it super easy to understand. Solving equations is a fundamental skill in mathematics, and once you get the hang of it, you'll be able to tackle all sorts of problems. We'll explore the different components and operations involved. So, grab your pens and paper, and let's get started! Understanding these concepts is essential. It's like learning the alphabet before you can read a book. Mastering equation solving opens doors to more advanced mathematical concepts and real-world applications. From calculating your finances to understanding scientific principles, the ability to solve equations is a powerful tool. The goal is to isolate the variable, which in this case is 'x,' on one side of the equation. We do this by applying inverse operations to both sides of the equation, ensuring that we maintain the equality. Remember, what you do to one side, you must do to the other. This ensures the equation remains balanced. It's like a seesaw; to keep it level, you must add or remove the same weight from both sides. This entire process allows us to unravel the equation and find the value of the unknown variable, 'x.' This is not just a mathematical exercise; it is also a fundamental skill applicable in countless scenarios.

Before we dive into solving the given equation, let's brush up on some key concepts. First, what does it mean to solve an equation? Essentially, we want to find the value of the variable (in this case, 'x') that makes the equation true. In other words, we want to find the number that, when substituted for 'x,' makes the left side of the equation equal to the right side. The core idea is to manipulate the equation, using mathematical operations, while keeping both sides balanced. Think of it like a puzzle where your goal is to isolate the variable. We use several fundamental operations like addition, subtraction, multiplication, and division. Each of these operations has an inverse operation. Inverse operations undo each other. For example, the inverse of addition is subtraction, and the inverse of multiplication is division. When simplifying expressions, remember to follow the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). Understanding these concepts is like having a toolkit ready for any mathematical challenge. Furthermore, always double-check your work, particularly by plugging the found solution back into the original equation to verify that it holds true. This is an essential practice in mathematics. It helps prevent errors and ensures you have the correct answer. The more you practice, the more comfortable and proficient you will become in solving these types of equations. Practice makes perfect, and with each equation you solve, your understanding and confidence will grow.

Step-by-Step Solution

Alright, let's roll up our sleeves and solve the equation: $-\frac{1}{2}\left(3 x-\frac{1}{3}\right)+x=4$. Here's how we'll do it, step by step:

Step 1: Distribute

First things first, we need to get rid of those parentheses. To do this, we'll distribute the $-\frac{1}{2}$ across the terms inside the parentheses. This means multiplying $-\frac{1}{2}$ by both $3x$ and $-\frac{1}{3}$.

So, $-\frac{1}{2} * 3x = -\frac{3}{2}x$ and $-\frac{1}{2} * -\frac{1}{3} = \frac{1}{6}$.

Our equation now looks like this: $- \frac{3}{2}x + \frac{1}{6} + x = 4$

Step 2: Combine Like Terms

Next, let's simplify things by combining like terms. In this case, we have two terms with 'x': $- \frac{3}{2}x$ and $+x$. Remember that $+x$ is the same as $+1x$.

So, $- \frac{3}{2}x + x = -\frac{3}{2}x + \frac{2}{2}x = -\frac{1}{2}x$

Our equation now simplifies to: $- \frac{1}{2}x + \frac{1}{6} = 4$

Step 3: Isolate the Variable

Now, let's isolate the 'x' term. First, we need to get rid of the constant term, $\frac{1}{6}$, on the left side. To do this, we'll subtract $\frac{1}{6}$ from both sides of the equation.

So, $\frac{1}{6} - \frac{1}{6} = 0$, and $4 - \frac{1}{6} = \frac{24}{6} - \frac{1}{6} = \frac{23}{6}$.

Our equation now becomes: $- \frac{1}{2}x = \frac{23}{6}$

Step 4: Solve for x

Finally, we need to get 'x' by itself. We have $-\frac{1}{2}x = \frac{23}{6}$. To solve for 'x,' we can multiply both sides of the equation by $-2$ (the reciprocal of $-\frac{1}{2}$). Doing this will cancel out the $-\frac{1}{2}$ on the left side.

So, $-2 * -\frac{1}{2}x = x$ and $-2 * \frac{23}{6} = -\frac{46}{6} = -\frac{23}{3}$

Therefore, $x = -\frac{23}{3}$

Verification

Always a good idea to check your solution. Let’s plug $x = -\frac{23}{3}$ back into the original equation to make sure it's correct.

$-\frac{1}{2}\left(3 * -\frac{23}{3} - \frac{1}{3}\right) + -\frac{23}{3} = 4$

Simplifying this, we get:

$-\frac{1}{2}\left(-23 - \frac{1}{3}\right) + -\frac{23}{3} = 4$

$-\frac{1}{2}\left(-\frac{69}{3} - \frac{1}{3}\right) - \frac{23}{3} = 4$

$-\frac{1}{2}\left(-\frac{70}{3}\right) - \frac{23}{3} = 4$

$\frac{70}{6} - \frac{23}{3} = 4$

$\frac{35}{3} - \frac{23}{3} = 4$

$\frac{12}{3} = 4$

$4 = 4$

Since both sides are equal, our solution $x = -\frac{23}{3}$ is correct!

Tips and Tricks for Success

Here are some helpful tips to make solving equations a breeze:

• Practice Regularly: The more equations you solve, the more comfortable you'll become. Consistency is key!
• Show Your Work: Writing down each step helps prevent errors and makes it easier to spot mistakes.
• Double-Check: Always verify your solution by plugging it back into the original equation. It's a lifesaver!
• Master the Basics: Make sure you understand the order of operations, fractions, and how to combine like terms.
• Don't Be Afraid to Ask: If you get stuck, don't hesitate to ask for help from a teacher, tutor, or friend. Everyone needs a little guidance sometimes.
• Break It Down: If an equation seems complex, break it down into smaller, manageable steps. This will make the process less overwhelming.
• Learn from Mistakes: Everyone makes mistakes. Instead of getting discouraged, view them as learning opportunities.

Conclusion

And there you have it! We've successfully solved the equation $-\frac{1}{2}\left(3 x-\frac{1}{3}\right)+x=4$ step-by-step. Remember, practice is key. Keep working through problems, and you'll become a pro in no time. Solving equations is a fundamental skill with broad applications. From basic arithmetic to advanced calculus, the ability to solve equations is an essential foundation. By understanding the principles we've covered today, you're well on your way to mastering this crucial skill. Embrace the process, enjoy the challenge, and keep practicing!

Hopefully, this breakdown has helped you understand the process better. Feel free to practice more equations to master this concept. Keep practicing, and you'll find that solving equations becomes easier and more enjoyable. Keep up the excellent work, and always remember the importance of practice and persistence in math. Keep up the great work! You got this!