Solve 2x + 5 = 13: A Step-by-Step Guide

Hey everyone! Let's dive into a classic algebra problem today. We're going to break down how to solve for x in the equation 2x + 5 = 13. Don't worry, even if algebra feels like a jumble of numbers and letters right now, we'll make it crystal clear. Think of these equations as puzzles, and we're the detectives cracking the code! Our goal is to isolate x on one side of the equation, revealing its true value. So, grab your thinking caps, and let’s get started!

Understanding the Basics: What Does it Mean to Solve for x?

Before we jump into the nitty-gritty, let's make sure we're all on the same page about what it means to "solve for x". When we say we want to solve for x, we mean we want to find the numerical value that x represents in this particular equation. In other words, we're looking for the number that, when multiplied by 2 and then added to 5, equals 13. To do this, we need to manipulate the equation using some basic algebraic principles. The golden rule of algebra is that whatever you do to one side of the equation, you must do to the other. This ensures that the equation remains balanced, like a scale. Imagine the equals sign (=) as the fulcrum of a scale; both sides must weigh the same to keep it level. We'll use this principle throughout our solution to keep things fair and square. The key here is to perform 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. We'll use these inverse operations to peel away the layers around x until we have it all by itself on one side of the equation. Think of it like unwrapping a present, one layer at a time, until you get to the gift inside – in this case, the value of x. So, with these foundational concepts in mind, let's roll up our sleeves and start solving!

Step 1: Isolating the Term with x

Okay, the first step in solving for x in the equation 2x + 5 = 13 is to isolate the term that contains x. In this case, that's the term 2x. Remember our golden rule? Whatever we do to one side, we do to the other. We want to get 2x by itself on the left side of the equation. Notice that we have a + 5 hanging out there with the 2x. To get rid of it, we need to perform the inverse operation of addition, which is subtraction. So, we're going to subtract 5 from both sides of the equation. This is crucial because it maintains the balance of the equation. If we only subtracted 5 from the left side, the equation would no longer be true. By subtracting from both sides, we're essentially saying, "We're taking away the same amount from both sides, so the balance remains undisturbed." Let's write this out mathematically: 2x + 5 - 5 = 13 - 5. On the left side, the + 5 and - 5 cancel each other out, leaving us with just 2x. On the right side, 13 - 5 equals 8. So, our equation now looks like this: 2x = 8. We've successfully isolated the term with x! We're one step closer to finding the value of x. This step is super important because it simplifies the equation and brings us closer to our goal. It's like clearing away the clutter on your desk so you can focus on the task at hand. Now that we have 2x by itself, we can move on to the next step, which involves getting rid of the coefficient attached to x. Keep your eyes peeled; we're almost there!

Step 2: Solving for x by Division

Alright, we've made it to the final step! We're currently at the equation 2x = 8. Remember, our ultimate goal is to get x all by itself on one side of the equation. Right now, x is being multiplied by 2. To undo this multiplication, we need to perform the inverse operation, which is division. Just like before, we need to divide both sides of the equation by the same number to maintain balance. In this case, we'll divide both sides by 2. This will isolate x and reveal its value. Let's write it out: (2x) / 2 = 8 / 2. On the left side, the 2 in the numerator and the 2 in the denominator cancel each other out, leaving us with just x. This is exactly what we wanted! On the right side, 8 divided by 2 equals 4. So, the equation simplifies to: x = 4. We've done it! We've successfully solved for x. This means that the value of x that makes the original equation true is 4. It's like finding the missing piece of a puzzle and slotting it into place. We now know the value that x represents in this equation. But wait, we're not quite done yet. It's always a good idea to check our work to make sure we haven't made any mistakes along the way. So, in the next section, we'll plug our solution back into the original equation to verify that it holds true.

Step 3: Verifying the Solution

Okay, guys, we've solved for x, and we think the answer is 4. But before we declare victory, it's super important to double-check our work. Think of it like proofreading a paper before you submit it – you want to make sure everything is perfect! To verify our solution, we're going to substitute x = 4 back into the original equation: 2x + 5 = 13. This is where the rubber meets the road. If our solution is correct, plugging in 4 for x should make the equation true. So, let's do it! Replacing x with 4, we get: 2(4) + 5 = 13. Now, we need to simplify the left side of the equation. First, we multiply 2 by 4, which gives us 8. So, the equation becomes: 8 + 5 = 13. Next, we add 8 and 5, which gives us 13. So, the left side of the equation simplifies to 13. Now, let's look at the entire equation: 13 = 13. This is a true statement! The left side of the equation equals the right side of the equation. This means that our solution, x = 4, is correct. We've verified our answer, and we can confidently say that we've solved the equation. Checking our work is a crucial step in any math problem. It helps us catch any errors we might have made and ensures that our solution is accurate. It's like having a safety net – it gives you peace of mind knowing that you've done everything correctly. So, always remember to verify your solutions whenever you can!

Conclusion: x = 4 is the Answer!

Woohoo! We did it! We successfully solved for x in the equation 2x + 5 = 13, and we found that x equals 4. We walked through each step of the process, from isolating the term with x to dividing both sides of the equation to finally verifying our solution. Remember, the key to solving algebraic equations is to use inverse operations and maintain balance. Whatever you do to one side of the equation, you must do to the other. This ensures that the equation remains true and that you arrive at the correct solution. Solving for x is a fundamental skill in algebra, and it's a skill that you'll use again and again in more advanced math courses. So, mastering these basic steps is super important. And don't forget the importance of verifying your solution! It's the final check that ensures your answer is correct. So, next time you encounter an equation like this, remember the steps we've covered, and you'll be well on your way to solving it with confidence. Keep practicing, and you'll become an algebra whiz in no time! Now go forth and conquer those equations!