Solving For X In The Equation 5x + 3 = 18 A Step-by-Step Guide

Let's dive into solving a basic algebraic equation! In this guide, we'll break down how to solve for 'x' in the equation 5x + 3 = 18. Whether you're brushing up on your algebra skills or tackling this for the first time, we'll make sure you understand each step. So, grab your pencil and paper, and let's get started!

Understanding the Basics

Before we jump into the solution, let's quickly review some fundamental concepts. In algebra, our goal is often to isolate a variable—in this case, 'x'—on one side of the equation. This means we want to get 'x' by itself so we can determine its value. To do this, we use inverse operations. Inverse operations are operations that undo each other. For example, addition and subtraction are inverse operations, and multiplication and division are inverse operations. Keeping this in mind will help us as we work through the equation. Remember guys, algebra might seem daunting at first, but with a clear understanding of the basics, it becomes much easier. Think of an equation like a balanced scale; whatever you do on one side, you must do on the other to keep it balanced. This principle is crucial when solving for 'x'. We're essentially performing operations to maintain the equality while isolating the variable. Don't worry if it doesn't click immediately. Practice makes perfect, and we're here to guide you through every step. We will also encounter terms and coefficients. A term is a single mathematical expression, which can be a constant, a variable, or a combination of both (like 5x or 3). The coefficient is the number multiplied by the variable (in 5x, the coefficient is 5). Recognizing these components will help you manipulate the equation more effectively. Also, keep an eye out for the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). While we won’t be using all of these operations in this specific equation, understanding the order is vital for more complex problems. Lastly, remember that the ultimate goal is clarity and accuracy. Show your work, double-check your steps, and don’t hesitate to go back and review if something doesn’t make sense. With these foundations in place, we are ready to tackle our equation and solve for ‘x’ with confidence.

Step 1: Isolating the Term with 'x'

Our first mission is to isolate the term that contains 'x', which in our equation 5x + 3 = 18, is '5x'. To do this, we need to get rid of the '+ 3' on the left side of the equation. Remember what we discussed about inverse operations? The inverse operation of addition is subtraction. So, to eliminate the '+ 3', we subtract 3 from both sides of the equation. It’s crucial to perform the same operation on both sides to maintain the balance. This gives us: 5x + 3 - 3 = 18 - 3. Simplifying this, we get 5x = 15. See how we’re one step closer to isolating 'x'? This step is a classic example of applying the golden rule of algebra: what you do to one side, you must do to the other. By subtracting 3 from both sides, we've effectively moved the constant term to the right side, paving the way for us to isolate 'x' completely. Now, you might be wondering why we subtract 3 specifically. Well, we chose 3 because it's the number being added to the term with 'x'. Our goal is to create a zero pair (+3 and -3 cancel each other out), leaving us with just '5x' on the left side. This technique is fundamental in algebra and will be used extensively in more complex equations. It is important to double-check your calculations at this stage. A small mistake here can throw off the entire solution. Ensure that you've correctly subtracted 3 from both sides and that you've simplified the equation accurately. Keep in mind that this step is not just about getting rid of the '+ 3'; it's about strategically manipulating the equation to reveal the value of 'x'. We're not just crunching numbers; we're solving a puzzle! And with each step, we're getting closer to the final piece. So, pat yourselves on the back – you've successfully isolated the term with 'x'. Let's move on to the next step and finish the job!

Step 2: Solving for 'x'

Now that we have 5x = 15, our next goal is to isolate 'x' completely. Currently, 'x' is being multiplied by 5. To undo this multiplication, we need to use the inverse operation, which is division. We'll divide both sides of the equation by 5. This gives us: 5x / 5 = 15 / 5. On the left side, 5x divided by 5 simplifies to just 'x', because 5 divided by 5 is 1. On the right side, 15 divided by 5 is 3. So, we have x = 3. That's it! We've solved for 'x'! This step showcases the power of inverse operations in algebra. By dividing both sides by the coefficient of 'x', we've effectively unwrapped 'x' and revealed its value. Think of it like peeling back the layers of an onion – each inverse operation gets us closer to the core, which is 'x'. It's awesome, right guys? Now, let's take a moment to appreciate what we've done here. We started with an equation that seemed to hide the value of 'x', and through a couple of strategic steps, we've uncovered it. This is the essence of algebra – using tools and techniques to solve for the unknown. It’s also important to note the elegance of this solution. By applying the principles of inverse operations and maintaining balance, we've arrived at a clear and concise answer. There are no shortcuts or tricks here; just solid algebraic thinking. Before we move on, let’s take a deep breath and savor this moment of triumph. We've successfully navigated the equation and found the value of 'x'. This feeling of accomplishment is what makes math so rewarding. Remember, each equation you solve is a victory – a step forward in your understanding and skill. So, congratulations! You've conquered this step, and you're well on your way to mastering algebraic equations. Let's move on to verifying our solution and ensuring that our answer is correct.

Step 3: Verifying the Solution

To make sure our solution x = 3 is correct, we need to plug it back into the original equation, 5x + 3 = 18. This process is called verification, and it's a crucial step in solving any equation. It helps us catch any mistakes we might have made along the way. So, let's substitute x with 3 in the original equation: 5 * (3) + 3 = 18. Now, we perform the multiplication first: 15 + 3 = 18. Then, we add: 18 = 18. The left side of the equation equals the right side, which means our solution is correct! High five! Verification is like the ultimate check-up for your solution. It’s where you put your answer to the test and see if it holds up. If the equation balances out, you know you've nailed it. If not, it's a sign to go back and review your steps. Think of it as a detective double-checking their work to make sure they've got the right suspect. It is also a great habit to develop because it builds confidence in your problem-solving abilities. Knowing that you've verified your answer gives you peace of mind and solidifies your understanding of the process. Plus, it's a fantastic way to avoid silly mistakes that can sometimes slip through. Let's consider what might happen if our solution didn't verify. Suppose we had made an error and gotten x = 4 as our answer. Plugging that into the original equation would give us 5 * (4) + 3 = 23, which is not equal to 18. This discrepancy would immediately alert us to the fact that something went wrong. The beauty of verification is that it provides a clear and objective way to assess our work. There's no guesswork involved; the numbers either balance out, or they don't. So, always make it a point to verify your solutions, guys. It's a small step that can make a big difference in your accuracy and confidence. And in this case, our solution checks out perfectly. We can confidently say that x = 3 is the correct answer to the equation 5x + 3 = 18. You've done an excellent job! Let's wrap things up with a quick summary of what we've learned.

Conclusion

In this step-by-step guide, we've successfully solved for 'x' in the equation 5x + 3 = 18. We started by understanding the basics of algebraic equations, including the concept of inverse operations. Then, we isolated the term with 'x' by subtracting 3 from both sides of the equation. Next, we solved for 'x' by dividing both sides by 5, which gave us x = 3. Finally, we verified our solution by plugging it back into the original equation, confirming that our answer was correct. Solving algebraic equations is a fundamental skill in mathematics, and mastering these steps will help you tackle more complex problems in the future. Remember, practice makes perfect, so keep working on different equations to strengthen your understanding. Guys, you've done a fantastic job following along, and I hope this guide has made the process clear and straightforward for you. Algebra can be challenging, but with the right approach and a bit of practice, you can conquer any equation that comes your way. The key takeaways from this exercise are the importance of inverse operations, maintaining balance in the equation, and the critical step of verification. These principles are not just applicable to this specific equation but are universal tools in the world of algebra. As you continue your mathematical journey, you'll encounter increasingly complex problems, but the foundational skills we've covered here will serve you well. Don't be afraid to make mistakes – they are valuable learning opportunities. Each error you encounter is a chance to deepen your understanding and refine your problem-solving techniques. And remember, there are plenty of resources available to support you, from textbooks and online tutorials to teachers and classmates. So, embrace the challenge, stay curious, and keep exploring the fascinating world of mathematics. You've already taken a significant step forward by working through this guide, and I'm confident that you'll continue to grow and excel in your mathematical endeavors. Keep up the great work, and never stop learning!