Solving 3x - 2 / 4 - 2x + 3 / 3 = 2/3 - X A Step-by-Step Guide
Are you struggling with linear equations? No worries, guys! You're in the right place. In this article, we're going to break down how to solve the linear equation 3x - 2 / 4 - 2x + 3 / 3 = 2/3 - x step by step. Linear equations might seem intimidating at first, but with a clear understanding of the process, you'll be solving them like a pro in no time. So, let's dive in and get started!
Understanding Linear Equations
Before we jump into the solution, let's quickly recap what a linear equation actually is. A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable. These equations are called "linear" because when graphed, they form a straight line. The general form of a linear equation is ax + b = c, where x is the variable, and a, b, and c are constants.
Linear equations are fundamental in mathematics and have applications across various fields, including physics, engineering, economics, and computer science. Mastering the art of solving them is crucial for anyone delving deeper into these areas. The key to solving linear equations lies in isolating the variable on one side of the equation. This is achieved by performing the same operations on both sides of the equation, ensuring the balance is maintained. We'll be using this principle extensively in our step-by-step guide below.
Step-by-Step Solution to 3x - 2 / 4 - 2x + 3 / 3 = 2/3 - x
Now, let’s tackle the equation at hand: 3x - 2 / 4 - 2x + 3 / 3 = 2/3 - x. This equation might look a bit complex with all the fractions, but don't worry, we'll break it down into manageable steps. Our goal is to isolate x on one side of the equation. To do this effectively, we'll follow a series of algebraic manipulations, ensuring we maintain the equation's balance at every step. Let's get started!
1. Clear the Fractions
The first step in solving this equation is to get rid of the fractions. Fractions can make equations look messy and complicated, so clearing them out simplifies the process significantly. To do this, we need to find the least common multiple (LCM) of the denominators. In our equation, the denominators are 4 and 3. The LCM of 4 and 3 is 12. Multiplying both sides of the equation by the LCM will eliminate the fractions. This is because multiplying each fraction by the LCM will result in the denominator canceling out, leaving us with whole numbers.
So, we multiply both sides of the equation by 12:
12 * (3x - 2 / 4 - 2x + 3 / 3) = 12 * (2/3 - x)
Now, distribute the 12 on both sides:
12 * (3x / 4) - 12 * (2 / 4) - 12 * (2x / 3) + 12 * (3 / 3) = 12 * (2 / 3) - 12 * x
Simplify each term:
9x - 6 - 8x + 12 = 8 - 12x
2. Combine Like Terms
Now that we've cleared the fractions, the equation looks much simpler. The next step is to combine like terms on each side of the equation. Like terms are terms that have the same variable raised to the same power. In our case, we have terms with x and constant terms. Combining like terms simplifies the equation further and makes it easier to isolate the variable.
On the left side, we have 9x and -8x, which can be combined to give us x. We also have -6 and +12, which combine to give us +6. So, the left side of the equation simplifies to x + 6.
x + 6 = 8 - 12x
3. Isolate the Variable Term
Our goal is to get all the terms with x on one side of the equation and the constant terms on the other side. To do this, we need to move the -12x term from the right side to the left side. We can achieve this by adding 12x to both sides of the equation. Remember, whatever we do to one side of the equation, we must do to the other to maintain the balance.
x + 6 + 12x = 8 - 12x + 12x
This simplifies to:
13x + 6 = 8
4. Isolate the Variable
Now we have 13x + 6 = 8. To isolate x, we need to get rid of the +6 on the left side. We can do this by subtracting 6 from both sides of the equation.
13x + 6 - 6 = 8 - 6
This simplifies to:
13x = 2
5. Solve for x
Finally, we're in the last step! We have 13x = 2. To solve for x, we need to divide both sides of the equation by 13. This will give us the value of x.
x = 2 / 13
So, the solution to the equation 3x - 2 / 4 - 2x + 3 / 3 = 2/3 - x is x = 2/13.
Common Mistakes to Avoid
Solving linear equations can be tricky, and it's easy to make mistakes along the way. Let's go over some common pitfalls to avoid. By being aware of these mistakes, you can improve your accuracy and problem-solving skills.
Forgetting to Distribute
One of the most common mistakes is forgetting to distribute when multiplying a number by an expression inside parentheses. For instance, if you have 2(x + 3), you need to multiply both the x and the 3 by 2, resulting in 2x + 6. Failing to distribute correctly can lead to significant errors in your solution. Always double-check your distribution steps to ensure you've multiplied each term inside the parentheses.
Incorrectly Combining Like Terms
Another frequent mistake is incorrectly combining like terms. Remember, you can only combine terms that have the same variable raised to the same power. For example, 3x and 2x are like terms and can be combined to give 5x. However, 3x and 2x² are not like terms and cannot be combined. Be careful to identify and combine only the correct terms to avoid errors.
Sign Errors
Sign errors are also very common, especially when dealing with negative numbers. Make sure to pay close attention to the signs when adding, subtracting, multiplying, and dividing. For example, subtracting a negative number is the same as adding a positive number, and vice versa. Double-checking your signs at each step can help prevent these types of errors.
Not Performing Operations on Both Sides
A fundamental rule of solving equations is that whatever operation you perform on one side, you must also perform on the other side. This maintains the balance of the equation. For example, if you add 5 to the left side, you must also add 5 to the right side. Failing to do this will lead to an incorrect solution. Always ensure you're applying the same operations to both sides of the equation.
Clearing Fractions Incorrectly
When clearing fractions, it’s crucial to multiply every term on both sides of the equation by the least common multiple (LCM). A common mistake is to multiply only some terms, leaving others untouched. This will disrupt the balance of the equation and lead to an incorrect answer. Double-check that you've multiplied every single term by the LCM to ensure the fractions are cleared correctly.
Tips for Mastering Linear Equations
Want to become a whiz at solving linear equations? Here are some tips and tricks to help you master the process and boost your confidence:
Practice Regularly
The golden rule of mathematics is practice makes perfect. The more you practice solving linear equations, the more comfortable and confident you'll become. Try solving a variety of problems, from simple to more complex, to reinforce your understanding. Regular practice will also help you identify patterns and develop a knack for solving equations efficiently.
Show Your Work
It might be tempting to do calculations in your head, but it's always a good idea to show your work step by step. Writing out each step makes it easier to spot mistakes and helps you understand the process more clearly. Plus, if you do make an error, it's much easier to trace back and find where you went wrong if you have a written record of your steps.
Check Your Answers
Always take the time to check your answers. Once you've found a solution, plug it back into the original equation to see if it holds true. If both sides of the equation are equal, you know your solution is correct. Checking your answers not only ensures accuracy but also helps reinforce your understanding of the equation-solving process.
Understand the Underlying Concepts
Memorizing steps can be helpful, but it's even more important to understand the underlying concepts. Know why you're performing each operation and how it affects the equation. A solid conceptual understanding will enable you to tackle a wider range of problems and adapt your approach when necessary.
Break Down Complex Problems
Complex linear equations can seem daunting, but they become much more manageable when you break them down into smaller, more manageable steps. Identify the key steps, such as clearing fractions, combining like terms, and isolating the variable, and tackle each step one at a time. This systematic approach will make even the most challenging equations solvable.
Use Online Resources
There are tons of fantastic online resources available to help you with linear equations. Websites like Khan Academy, Mathway, and Symbolab offer lessons, practice problems, and step-by-step solutions. These resources can be invaluable for reinforcing your understanding and getting extra practice.
Seek Help When Needed
Don't hesitate to ask for help if you're struggling. Talk to your teacher, a tutor, or a classmate. Explaining your difficulties can often help you clarify your understanding, and getting a different perspective can provide valuable insights. Remember, everyone struggles sometimes, and seeking help is a sign of strength, not weakness.
Conclusion
Solving linear equations is a crucial skill in mathematics, and with the right approach, it's totally achievable. In this article, we've walked through a step-by-step solution to the equation 3x - 2 / 4 - 2x + 3 / 3 = 2/3 - x, highlighting the key steps and common mistakes to avoid. By following these guidelines and practicing regularly, you'll be solving linear equations with confidence in no time. So keep practicing, stay patient, and you'll become a linear equation master! Remember, guys, math is a journey, not a destination. Enjoy the process and celebrate your progress along the way!