Solving The Equation 6x + 3x - 5 = 18 + 4: A Step-by-Step Guide

Hey guys! Ever stumbled upon an equation that looks like a jumbled mess of numbers and letters? Don't worry, we've all been there. Today, we're going to break down the equation 6x + 3x - 5 = 18 + 4 step-by-step, making it super easy to understand and solve. So, grab your pencils and let's dive into the world of algebra!

Understanding the Basics of Algebraic Equations

Before we jump into solving this specific equation, let's quickly refresh the fundamental concepts of algebraic equations. In essence, an algebraic equation is a mathematical statement asserting that two expressions are equal. These expressions often contain variables, which are symbols (usually letters like 'x', 'y', or 'z') representing unknown values. The goal of solving an equation is to find the value(s) of the variable(s) that make the equation true. To achieve this, we employ various algebraic operations, ensuring that we maintain the balance of the equation—what we do on one side, we must also do on the other. This principle of balance is the cornerstone of equation-solving, allowing us to manipulate and simplify the equation until we isolate the variable and determine its value. Understanding this foundational concept is crucial as it underpins all algebraic problem-solving strategies, making the process of tackling complex equations much more manageable and intuitive. So, always remember, balance is key! When faced with a challenging equation, breaking it down into smaller, manageable steps, while keeping this core principle in mind, will lead you to the solution.

Key Components of an Equation

Let's break down the anatomy of an equation. You've got your variables (like 'x'), which are the unknowns we're trying to find. Then, there are coefficients, the numbers that multiply the variables (like the '6' and '3' in '6x' and '3x'). We also have constants, which are just plain numbers (like '-5', '18', and '4'). The equals sign (=) is the heart of the equation, showing that both sides have the same value. Think of it like a balancing scale – whatever you do to one side, you gotta do to the other to keep it level. This principle of maintaining balance is crucial when solving equations. For instance, if you add a number to one side, you must add the same number to the other side to maintain the equality. Similarly, if you multiply one side by a certain value, you need to multiply the other side by the same value. This ensures that the equation remains true and that you are progressing towards the correct solution. Understanding these components – variables, coefficients, constants, and the equals sign – is the first step in mastering the art of equation solving. It lays the groundwork for tackling more complex problems and developing a strong foundation in algebra.

Step-by-Step Solution to 6x + 3x - 5 = 18 + 4

Now, let's get down to business and solve our equation. We'll take it one step at a time, so you can see exactly how it's done.

Step 1: Simplify Both Sides of the Equation

First things first, let's simplify both sides of the equation. On the left side, we have 6x + 3x - 5. We can combine the 'x' terms since they're like terms. Think of it like having 6 apples plus 3 apples, which gives you 9 apples. So, 6x + 3x becomes 9x. Now our left side is 9x - 5. On the right side, we have 18 + 4, which is a simple addition. 18 + 4 equals 22. So, our simplified equation looks like this: 9x - 5 = 22. Simplifying both sides is a crucial first step in solving any equation because it makes the equation easier to work with. By combining like terms and performing basic arithmetic, we reduce the complexity and make the subsequent steps more straightforward. This approach is not just about finding the solution; it's also about making the process more manageable and less prone to errors. Always aim to simplify as much as possible at the beginning, as this will set you up for success in the later stages of solving the equation. Remember, a clean and simple equation is much easier to navigate!

Step 2: Isolate the Variable Term

Our next goal is to get the term with 'x' (which is 9x) all by itself on one side of the equation. To do this, we need to get rid of the '- 5' on the left side. Remember the balancing scale? We need to do the same thing to both sides to keep it balanced. The opposite of subtracting 5 is adding 5, so we'll add 5 to both sides of the equation. This gives us: 9x - 5 + 5 = 22 + 5. On the left side, the '- 5' and '+ 5' cancel each other out, leaving us with just 9x. On the right side, 22 + 5 equals 27. So now we have: 9x = 27. Isolating the variable term is a pivotal step because it brings us closer to determining the value of 'x'. By strategically adding or subtracting values from both sides, we're essentially peeling away the layers surrounding the variable until it stands alone. This process requires a clear understanding of inverse operations – knowing that addition undoes subtraction, and vice versa. It's like a mathematical dance where we carefully maneuver terms to create a clear path to the solution. Mastering this step is crucial for solving more complex equations, as it's a fundamental technique used across various algebraic problems. Remember, the key is to focus on the variable term and methodically eliminate any other constants or coefficients that are attached to it.

Step 3: Solve for x

We're almost there! Now we have 9x = 27. This means 9 times 'x' equals 27. To find out what 'x' is, we need to undo the multiplication. The opposite of multiplying by 9 is dividing by 9, so we'll divide both sides of the equation by 9. This gives us: 9x / 9 = 27 / 9. On the left side, the '9's cancel each other out, leaving us with just 'x'. On the right side, 27 divided by 9 is 3. So, we finally have: x = 3. And that's our solution! Solving for 'x' is the culmination of all our efforts, the moment where we finally unveil the unknown value. It's the final piece of the puzzle, the answer we've been working towards throughout the entire process. This step typically involves performing the inverse operation of whatever is being done to the variable – whether it's multiplication, division, addition, or subtraction. It requires a precise application of mathematical principles, ensuring that we maintain the balance of the equation and arrive at the correct solution. Once we isolate 'x' and determine its value, we've essentially cracked the code, solving the equation and demonstrating our understanding of algebraic concepts. This final step is not just about getting the answer; it's about solidifying our grasp of the entire equation-solving process.

The Answer

So, the solution to the equation 6x + 3x - 5 = 18 + 4 is x = 3. This corresponds to option A.

Checking Your Work

It's always a good idea to check your answer to make sure it's correct. To do this, we'll plug our solution (x = 3) back into the original equation and see if it makes the equation true. Our original equation was 6x + 3x - 5 = 18 + 4. Let's substitute '3' for 'x':

• 6(3) + 3(3) - 5 = 18 + 4
• 18 + 9 - 5 = 22
• 27 - 5 = 22
• 22 = 22

The left side equals the right side, so our solution is correct! Checking your work is an essential practice in mathematics, serving as a safety net to catch any potential errors. It's like proofreading a document before submitting it, ensuring that everything is accurate and makes sense. By substituting your solution back into the original equation, you're essentially putting your answer to the test, verifying that it satisfies the conditions of the problem. This not only confirms the correctness of your solution but also reinforces your understanding of the equation itself. It's a valuable habit to develop, as it builds confidence in your problem-solving abilities and minimizes the risk of submitting incorrect answers. So, always take the time to check your work – it's a small investment that can yield significant returns.

Tips for Solving Algebraic Equations

Here are a few extra tips to help you become a pro at solving algebraic equations:

1. Simplify First: Always simplify both sides of the equation as much as possible before you start moving terms around.
2. Isolate the Variable: Your goal is to get the variable by itself on one side of the equation. Use inverse operations to move terms.
3. Check Your Work: Plug your solution back into the original equation to make sure it works.
4. Practice Makes Perfect: The more you practice, the better you'll get at solving equations. Don't be afraid to make mistakes – they're part of the learning process!

Conclusion

Solving equations might seem tricky at first, but with practice and a step-by-step approach, you can conquer any equation that comes your way. Remember to simplify, isolate the variable, and always check your work. You've got this! Keep practicing, and soon you'll be an algebra whiz. And hey, if you ever get stuck, don't hesitate to ask for help. We're all in this together!