Solving 2x + 5 = 15 Step-by-Step Solution

Hey guys! Ever get stuck on a math problem and feel like you're staring at a jumbled mess of numbers and symbols? Don't worry, we've all been there. Today, we're going to break down a common type of equation and show you how to solve it step-by-step. Let's dive into the equation 2x + 5 = 15 and find out what 'x' really is. Understanding the fundamentals of algebra, especially how to isolate variables in equations, is crucial for anyone looking to enhance their math skills. This equation, 2x + 5 = 15, might seem daunting at first glance, but it's actually quite straightforward once you break it down into manageable steps. Our goal here is to demystify the process and provide you with a clear, concise method to solve similar algebraic equations. So, grab your pencils, and let's get started on this mathematical journey together! Whether you're a student tackling homework, a professional needing a refresher, or just someone curious about algebra, this guide is tailored to help you grasp the concept and confidently solve equations like this one. We will focus on the underlying principles of equation solving, such as maintaining balance and performing inverse operations. These are not just rules to memorize, but concepts that will build a strong foundation for more advanced mathematics. By understanding why we perform each step, you'll be better equipped to tackle any algebraic challenge that comes your way. Remember, math is not just about getting the right answer; it's about understanding the process and developing logical thinking skills. This step-by-step guide will not only show you how to solve 2x + 5 = 15, but also empower you to approach other equations with confidence and clarity. So, let's embark on this adventure together and unlock the secrets of algebra!

Step 1: Understanding the Equation

Before we jump into solving, let's make sure we understand what the equation 2x + 5 = 15 is telling us. In this equation, 'x' is the unknown value we're trying to find. The number '2' is multiplying 'x' (2x means 2 times x), and then we're adding '5' to that result. The equals sign (=) tells us that the entire expression on the left side (2x + 5) has the same value as the number on the right side (15). Think of an equation like a balanced scale. The equals sign is the center point, and both sides must weigh the same to keep the scale balanced. Our job is to figure out what value of 'x' will make the left side equal to 15, thus keeping the scale balanced. To truly grasp this, let's consider what each component of the equation represents in the real world. Imagine you have a certain number of boxes, each containing an unknown quantity of items represented by 'x'. You have two such boxes (hence, 2x). Additionally, you have 5 individual items. In total, you have 15 items. The equation 2x + 5 = 15, therefore, is a mathematical representation of this situation. Our mission is to find out how many items are in each box. This understanding not only makes the equation more relatable but also helps in visualizing the steps we take to solve it. When we talk about maintaining the balance of the equation, it’s like ensuring the total number of items on both sides remains equal, no matter what operations we perform. By visualizing the equation in this way, the abstract symbols become more concrete, making the problem easier to approach and solve. This foundational understanding is key to successfully navigating more complex algebraic equations in the future.

Step 2: Isolating the Term with 'x'

The main goal in solving any equation is to get the variable (in this case, 'x') by itself on one side of the equation. This is called isolating the variable. To do this, we need to undo any operations that are being performed on 'x'. In our equation, 2x + 5 = 15, we see that 'x' is being multiplied by 2 and then 5 is being added. We need to undo these operations in reverse order. The first operation we'll undo is the addition of 5. To do this, we'll subtract 5 from both sides of the equation. Remember the balanced scale analogy? Whatever we do to one side of the equation, we must do to the other side to keep it balanced. So, we subtract 5 from both sides: 2x + 5 - 5 = 15 - 5. This simplifies to 2x = 10. Now, we've successfully isolated the term with 'x' (2x) on the left side. We've removed the addition, bringing us closer to finding the value of 'x' itself. Understanding why we perform this subtraction is crucial. We are using the concept of inverse operations. Addition and subtraction are inverse operations, meaning they undo each other. By subtracting 5 from both sides, we effectively cancel out the +5 on the left side, leaving us with just the term containing 'x'. This is a fundamental principle in algebra, and it’s the key to isolating variables in equations. The importance of maintaining balance cannot be overstated. Every operation we perform must be applied equally to both sides of the equation. This ensures that the equality holds true throughout the solving process. Failing to do so would disrupt the balance and lead to an incorrect solution. So, always remember, whatever you do to one side, you must do to the other. This principle is not just a rule, but a reflection of the fundamental nature of mathematical equality.

Step 3: Solving for 'x'

Now that we have 2x = 10, we're just one step away from finding the value of 'x'. Remember, 2x means 2 times x. So, to isolate 'x', we need to undo the multiplication. The inverse operation of multiplication is division. Therefore, we'll divide both sides of the equation by 2. Again, we need to do the same thing to both sides to maintain the balance. So, we have 2x / 2 = 10 / 2. This simplifies to x = 5. And there you have it! We've found the solution to the equation. The value of 'x' that makes the equation 2x + 5 = 15 true is 5. Let's break down the reasoning behind this step a bit further. Dividing both sides by 2 effectively cancels out the multiplication by 2 on the left side. Just as subtraction cancels out addition, division cancels out multiplication. This inverse operation is the key to isolating 'x'. The result, x = 5, tells us that if we substitute 5 for 'x' in the original equation, the equation will hold true. This is a crucial step in verifying our solution, which we'll discuss in the next section. Understanding this process of using inverse operations is fundamental to solving algebraic equations. It's not just about following steps; it's about understanding why those steps work. By grasping the underlying principles, you can confidently apply these techniques to a wide range of equations. Remember, the goal is to isolate the variable, and inverse operations are our tools to achieve that. Each step we take brings us closer to that goal, building our understanding and confidence along the way.

Step 4: Verifying the Solution

It's always a good idea to check our work and make sure our solution is correct. To verify our solution, we substitute the value we found for 'x' (which is 5) back into the original equation: 2x + 5 = 15. Replacing 'x' with 5, we get 2(5) + 5 = 15. Now, let's simplify the left side of the equation. First, we multiply 2 by 5, which gives us 10. So, the equation becomes 10 + 5 = 15. Finally, we add 10 and 5, which equals 15. This gives us 15 = 15, which is a true statement. Since the left side of the equation equals the right side when we substitute x = 5, we can be confident that our solution is correct. Verifying the solution is a critical step in the problem-solving process. It's like the final seal of approval, confirming that our calculations and reasoning are accurate. It not only ensures that we have the correct answer but also reinforces our understanding of the equation and the steps we took to solve it. Think of it as a safety net; it catches any potential errors and prevents us from moving forward with an incorrect solution. This step also highlights the importance of precision and attention to detail in mathematics. A small mistake in the process can lead to a wrong answer, but by verifying our solution, we can identify and correct these errors. Moreover, the verification process enhances our problem-solving skills. It encourages us to think critically about our work and develop a habit of checking our answers. This habit is invaluable not only in mathematics but also in other areas of life. So, always make sure to verify your solutions, and you'll be well on your way to mastering algebra.

Conclusion

Awesome! You've successfully solved the equation 2x + 5 = 15. By following these steps, you found that x = 5. Remember, the key is to understand the equation, isolate the variable using inverse operations, and always verify your solution. Math can seem tricky sometimes, but with a little practice and a step-by-step approach, you can conquer any equation! This journey through solving 2x + 5 = 15 has equipped you with valuable skills and insights into the world of algebra. You've learned how to approach an equation systematically, break it down into manageable steps, and apply the principles of inverse operations and balance. But more than just learning a specific solution, you've gained a deeper understanding of the problem-solving process itself. The skills you've acquired here are transferable to a wide range of mathematical challenges. Whether you're tackling more complex algebraic equations, geometry problems, or even real-world applications of math, the foundational knowledge you've built will serve you well. Remember, mathematics is not just about numbers and symbols; it's about logical thinking, problem-solving, and critical analysis. By embracing these skills, you can unlock a world of opportunities and possibilities. So, keep practicing, keep exploring, and never stop asking questions. The more you engage with math, the more confident and capable you'll become. And remember, every problem you solve is a step forward on your journey to mathematical mastery. Congratulations on your success, and keep up the great work!