Solving Equations: A Step-by-Step Guide

Hey everyone! Today, we're diving into the world of equations. Specifically, we're going to tackle the equation: $\frac{2x + 6}{2} = 2 - \frac{x}{3}$. Don't worry if equations seem intimidating at first; we'll break it down step-by-step to make sure it's crystal clear. Equations are fundamental in math, and understanding how to solve them opens doors to a whole bunch of other concepts. This guide will not only show you how to solve this specific equation but also equip you with the general skills to solve a wide variety of similar problems. So, grab your pencils and let's get started! We're going to make this a fun and rewarding journey, so you'll be a pro at solving equations in no time. Think of it like learning a new language – with each step, you become more fluent. Let's get into the nitty-gritty of how to solve this equation, ensuring we understand each part clearly.

First and foremost, understanding the basics is key. An equation is simply a mathematical statement that asserts the equality of two expressions. It's like a balance scale; what you do to one side, you must do to the other to keep it balanced. In our equation, we have 2x + 6 / 2 on one side and 2 - x / 3 on the other. Our primary goal is to isolate the variable x on one side of the equation. To do this, we'll use a series of operations: addition, subtraction, multiplication, and division. The key is to perform these operations in a way that simplifies the equation, bit by bit, until x is all by itself. It is important to keep the equation balanced throughout this process. Now, what exactly does 'isolating the variable' mean? It means getting x by itself on one side of the equal sign, with a numerical value on the other side. The final form we are aiming for is x = some number. We will meticulously step-by-step arrive at the solution. So let's get into it!

Step-by-Step Solution

Alright, let's roll up our sleeves and solve this equation together. We'll break down each step so you can follow along easily and fully grasp the process. This is where the real fun begins. We will take care with each individual step.

Step 1: Simplify the Fractions

Let's start by simplifying the fractions in our equation. Our equation is: $\frac{2x + 6}{2} = 2 - \frac{x}{3}$. The fraction on the left can be simplified. We have a 2x + 6 in the numerator and a 2 in the denominator. You can divide both terms in the numerator by the denominator: 2x / 2 = x and 6 / 2 = 3. That means the left side of our equation becomes x + 3. On the right side, we have 2 - x / 3. This side is already in its simplest form for this step. So, our equation now looks like this: x + 3 = 2 - x / 3. See, we've already made it a bit easier to handle. Always try to simplify any fractions or expressions first, as it significantly streamlines the subsequent steps. It reduces the complexity and minimizes the chance of errors. Remember that the goal is always to work towards isolating x, and the simplification process is a crucial part of achieving that goal. Remember to do each step with care. Think about the mathematics you are using to help solve for x.

Now that we've simplified the fractions as far as we can at this point, the next step is to focus on eliminating any remaining fractions. This is where the multiplication comes in. The more fractions that we get rid of, the easier it gets to solve.

Step 2: Eliminate Fractions

Next up, we're going to eliminate the fraction on the right side of the equation x + 3 = 2 - x / 3. To get rid of the fraction, we'll multiply every term in the equation by the least common multiple (LCM) of the denominators. In our case, the only denominator we need to consider is 3. So, we'll multiply every term by 3. Let's see what happens: 3 * (x) = 3x. Then, 3 * (3) = 9. On the right side, 3 * (2) = 6, and 3 * (x/3) = x. The new equation becomes: 3x + 9 = 6 - x. Do you see how we have gotten rid of the fractions? Multiplying by the LCM is a powerful technique. It simplifies the equation by getting rid of fractions, making it easier to isolate the variable. Always make sure to multiply every single term by the LCM; otherwise, your equation will be unbalanced, and you'll get the wrong answer. This step is crucial in streamlining the equation. The removal of fractions prepares the ground for the next stages where we'll bring all the x terms together and isolate x. By eliminating the fractions, we've cleared the way for the final steps to solve for x. Pay close attention to the arithmetic here, ensuring that each term is correctly multiplied.

Always be mindful of the rules of arithmetic when performing these operations to maintain the integrity of your equation. So, we've successfully eliminated the fraction, and we now have a much simpler equation to work with. Great job!

Step 3: Gather x Terms

Now it's time to gather all the terms with x on one side of the equation. Our current equation is 3x + 9 = 6 - x. We want to get all the x terms together. To do this, let's add x to both sides of the equation: 3x + x + 9 = 6 - x + x. That simplifies to 4x + 9 = 6. By adding x to both sides, we've eliminated the x term from the right side, and we've brought it over to the left side. This is one of the fundamental strategies in solving equations: getting all like terms together. So, we need to make sure that any x terms are on the same side. Now that we've gathered the x terms, the next step is to isolate x. We're getting closer to our goal: x = some number. The next step is to isolate x on one side of the equation.

We are getting closer to solving for x. So, we have gathered all our x terms to the same side, which brings us to the next crucial step: isolating x to solve for the solution. The next step is to move any numbers to the other side. You're doing awesome!

Step 4: Isolate x

Almost there! Our equation is currently 4x + 9 = 6. To isolate x, we need to get rid of the + 9 on the left side. We do this by subtracting 9 from both sides: 4x + 9 - 9 = 6 - 9. This simplifies to 4x = -3. We're almost there! Now the final step is to get x all by itself. By subtracting nine from both sides, we've cleared the constant term from the left side, inching us closer to the solution. What we're doing here is using the inverse operation to isolate the variable. Remember that to keep the equation balanced, what you do to one side must be done to the other. This step is about getting x alone. Make sure to carefully perform the subtraction on both sides to avoid errors. Now that we've isolated the term with x, we're ready for the final step to get the value of x.

Now that we've simplified the equation to 4x = -3, we're ready for the final step.

Step 5: Solve for x

To solve for x, we need to get x completely by itself. The equation we have is 4x = -3. Currently, x is being multiplied by 4. To isolate x, we do the opposite: we divide both sides of the equation by 4. So, 4x / 4 = -3 / 4. This gives us x = -3/4. Therefore, the solution to the equation is x = -3/4. Congratulations! You've solved the equation! The solution to this equation is x = -3/4, and you now know how to find the solution. By dividing both sides by 4, we've completely isolated x and found its value. This final step is the culmination of all our previous efforts. Always double-check your final answer by substituting the value of x back into the original equation to make sure it works. This step is to solve for x, and we did it! You now know how to work through an equation to solve for x. You did an amazing job!

Conclusion

Alright, folks, we've successfully navigated the equation-solving process step-by-step. From simplifying fractions to isolating x, we've covered it all. Equations might seem tricky at first, but with the right approach and practice, they become much more manageable. Remember the key takeaways: always simplify, eliminate fractions, gather like terms, and isolate the variable. Practice these steps, and you'll become a pro in no time. Keep in mind that solving equations is like building a puzzle; each step brings you closer to the complete picture. Don't be afraid to make mistakes – they're a great way to learn and improve. Keep practicing, and you'll find that solving equations becomes easier and more intuitive with time. So, keep up the great work, and happy equation solving!