Solving The Equation (3/7)x + (0/7) - 2x = 4 A Step-by-Step Guide

Hey guys! Today, we're diving into the exciting world of algebra to tackle the equation (3/7)x + (0/7) - 2x = 4. Don't worry if it looks a bit intimidating at first glance; we're going to break it down step by step, so you'll be solving equations like a pro in no time. Math can be like a puzzle, and each step we take is like fitting a piece into place. So, let's get started and solve this equation together! We'll make sure to explain every step clearly, so you can follow along easily and understand the underlying principles at play.

Understanding the Equation

Before we jump into solving, let's take a moment to understand what we're dealing with. The equation (3/7)x + (0/7) - 2x = 4 might look complex, but it's made up of simple components. We have terms involving 'x', which is our variable, and constant terms, which are just numbers. Our goal is to isolate 'x' on one side of the equation to find its value. Remember, equations are like a balancing scale; whatever we do on one side, we must also do on the other to keep it balanced. This principle is key to solving any algebraic equation. We need to carefully manipulate the equation, using mathematical operations, to gradually get 'x' by itself. This might involve combining like terms, adding or subtracting values, or multiplying or dividing both sides. Each step is a deliberate action aimed at simplifying the equation and bringing us closer to the solution.

Simplifying the Equation

The first thing we can do to simplify the equation is to deal with the term (0/7). Anything divided by zero is zero, so we can rewrite the equation as (3/7)x - 2x = 4. This immediately makes the equation cleaner and easier to work with. Next, we need to combine the terms that involve 'x'. We have (3/7)x and -2x. To combine these, we need a common denominator. We can rewrite -2x as (-14/7)x. So, the equation becomes (3/7)x + (-14/7)x = 4. Now, we can add the coefficients (the numbers in front of 'x') together. 3/7 plus -14/7 equals -11/7. Thus, our equation simplifies to (-11/7)x = 4. This step is crucial because it reduces the number of terms and makes the equation much more manageable. By combining like terms, we're essentially grouping together the 'x' terms, which helps us isolate 'x' more effectively. This simplification process is a fundamental technique in algebra, allowing us to tackle more complex equations by breaking them down into simpler components.

Solving for x

Now that we have (-11/7)x = 4, we're in the home stretch! To isolate 'x', we need to get rid of the fraction (-11/7) that's multiplying it. We can do this by multiplying both sides of the equation by the reciprocal of -11/7, which is -7/11. Remember, whatever we do to one side of the equation, we must do to the other to maintain balance. So, we multiply both sides by -7/11: (-7/11) * (-11/7)x = 4 * (-7/11). On the left side, the fractions cancel each other out, leaving us with just 'x'. On the right side, we have 4 multiplied by -7/11, which equals -28/11. Therefore, we find that x = -28/11. This step demonstrates a core principle of solving equations: using inverse operations to isolate the variable. Multiplication and division are inverse operations, so by multiplying by the reciprocal, we effectively "undo" the multiplication by -11/7. This technique is widely used in algebra and is essential for solving a wide range of equations.

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 = -28/11 back into the original equation: (3/7)x + (0/7) - 2x = 4. Plugging in -28/11 for 'x', we get (3/7)(-28/11) + (0/7) - 2(-28/11) = 4. Let's simplify this. (3/7)(-28/11) equals -12/11, and -2(-28/11) equals 56/11. So, the equation becomes -12/11 + 56/11 = 4. Adding the fractions, we get 44/11, which simplifies to 4. So, 4 = 4, which means our solution is correct! Checking our solution is a crucial step because it helps us catch any potential errors in our calculations. By substituting the value back into the original equation, we can verify that it satisfies the equation and that we haven't made any algebraic mistakes. This step provides confidence in our answer and reinforces the importance of accuracy in mathematical problem-solving.

Alternative Methods for Solving

While we solved the equation by simplifying and isolating 'x', there are other methods we could have used. For example, we could have multiplied the entire equation by 7 at the beginning to eliminate the fractions. This would give us 3x - 14x = 28, which simplifies to -11x = 28. Then, we could divide both sides by -11 to get x = -28/11, the same answer we found earlier. This alternative method highlights the flexibility in solving algebraic equations. There often isn't just one "right" way to approach a problem; rather, there are multiple paths to the solution. The key is to choose a method that you understand and that feels comfortable for you. By exploring different approaches, you can develop a deeper understanding of the underlying mathematical principles and become a more confident problem-solver.

Tips and Tricks for Equation Solving

Solving equations is a fundamental skill in mathematics, and there are a few tips and tricks that can make the process easier. First, always simplify the equation as much as possible before you start isolating the variable. This might involve combining like terms, distributing, or clearing fractions or decimals. Second, remember to perform the same operation on both sides of the equation to maintain balance. Third, check your solution by substituting it back into the original equation. And finally, practice makes perfect! The more equations you solve, the better you'll become at recognizing patterns and applying the appropriate techniques. Equation solving is like learning a new language; the more you practice, the more fluent you become. By mastering these skills, you'll unlock new levels of mathematical understanding and be able to tackle more complex problems with confidence.

Common Mistakes to Avoid

When solving equations, it's easy to make mistakes if you're not careful. One common mistake is forgetting to distribute a number or sign correctly. For example, if you have -2(x + 3), you need to distribute the -2 to both the 'x' and the 3, resulting in -2x - 6, not -2x + 6. Another common mistake is combining unlike terms. You can only add or subtract terms that have the same variable and exponent. For example, you can't combine 2x and 3x², because they have different exponents. A third common mistake is forgetting to perform the same operation on both sides of the equation. Remember, the equation is like a balance, and whatever you do to one side, you must do to the other. Being aware of these common mistakes can help you avoid them and solve equations more accurately. It's also helpful to show your work clearly, so you can easily identify any errors you might have made. By paying attention to detail and developing good problem-solving habits, you can minimize the chances of making mistakes and maximize your success in algebra.

Real-World Applications of Equation Solving

Equation solving isn't just an abstract mathematical concept; it has countless real-world applications. From calculating the trajectory of a rocket to determining the optimal dosage of a medication, equations are used in a wide variety of fields. In physics, equations are used to describe the motion of objects, the behavior of electricity and magnetism, and the properties of light and sound. In engineering, equations are used to design bridges, buildings, and machines. In economics, equations are used to model supply and demand, predict market trends, and analyze financial data. Even in everyday life, we use equations without realizing it, such as when we calculate the tip at a restaurant or figure out how much paint we need to cover a wall. Understanding how to solve equations is a valuable skill that can help you in many areas of life and work. It's a fundamental tool for problem-solving and critical thinking, and it empowers you to analyze and understand the world around you in a more quantitative way. By mastering equation solving, you're not just learning a mathematical technique; you're developing a valuable life skill.

Conclusion

So, there you have it! We've successfully solved the equation (3/7)x + (0/7) - 2x = 4, and we found that x = -28/11. Remember, the key to solving equations is to simplify, isolate the variable, and check your solution. With practice and a little bit of patience, you'll be able to tackle even the most challenging equations. Keep practicing, keep exploring, and most importantly, keep having fun with math! Math is more than just numbers and symbols; it's a powerful tool for understanding the world and solving problems. By embracing the challenges and celebrating the successes, you can develop a strong foundation in mathematics and unlock your full potential. And remember, every equation you solve is a step forward on your mathematical journey. So, keep going, and never stop learning!