Solving 2x + 3 = 15 A Step-by-Step Guide

Let's dive into the world of algebra, guys! Today, we're tackling a classic linear equation: 2x + 3 = 15. This might seem intimidating at first, but trust me, with a few simple steps, we can crack this code and find the value of 'x'. Think of it like a puzzle – we're given the pieces, and our job is to arrange them to reveal the solution. This equation is a fundamental concept in mathematics, and mastering it will open doors to more complex problem-solving in the future. We'll break down each step, explaining the logic behind the math so you not only get the answer but also understand the process. Understanding the process is key because it allows you to apply these principles to a wide variety of algebraic problems. The ability to solve linear equations is crucial not only for success in math class but also for various real-world applications, from calculating budgets to understanding scientific formulas. So, grab your pencils, and let's embark on this mathematical adventure together! We'll go from the basics to some more nuanced techniques, ensuring you're well-equipped to tackle any linear equation that comes your way. Remember, math isn't about memorization; it's about understanding the underlying principles and applying them creatively.

The Foundation: Understanding Linear Equations

Before we jump into solving 2x + 3 = 15, let's make sure we're all on the same page about what a linear equation actually is. In simple terms, a linear equation is an algebraic equation where the highest power of the variable (in this case, 'x') is 1. This means we won't see any x² or x³ terms. Linear equations represent a straight line when graphed, hence the name “linear.” The general form of a linear equation is ax + b = c, where 'a', 'b', and 'c' are constants (numbers), and 'x' is the variable we're trying to find. In our equation, 2x + 3 = 15, 'a' is 2, 'b' is 3, and 'c' is 15. Understanding this foundation is crucial because it provides the framework for our problem-solving strategy. Each part of the equation plays a specific role, and recognizing these roles helps us manipulate the equation to isolate 'x'. The beauty of linear equations lies in their predictability – they follow a set of rules that, once mastered, make solving them a breeze. We'll explore these rules in detail as we move through the solution process. Thinking about equations in this way, as a balancing act, can make the process more intuitive and less daunting. So, let's get ready to master the art of balancing equations and uncovering the hidden values of our variables!

Step-by-Step Solution: Cracking the Code of 2x + 3 = 15

Now, let's get down to business and solve our equation: 2x + 3 = 15. The core strategy here is to isolate 'x' on one side of the equation. We do this by performing the same operations on both sides, maintaining the balance. It's like a scale – whatever we do to one side, we must do to the other to keep it level. This principle is the cornerstone of algebraic manipulation. Our first step is to get rid of the '+ 3' on the left side. To do this, we subtract 3 from both sides of the equation. This gives us:

2x + 3 - 3 = 15 - 3

Simplifying this, we get:

2x = 12

Great! We're one step closer. Now, we have 2x = 12. This means 2 times 'x' equals 12. To isolate 'x', we need to undo the multiplication. The opposite of multiplication is division, so we divide both sides of the equation by 2:

2x / 2 = 12 / 2

This simplifies to:

x = 6

And there you have it! We've found our solution: x = 6. To be absolutely sure, we can substitute this value back into the original equation to check our work. This process of verification is a crucial step in problem-solving, ensuring accuracy and building confidence in our solutions. It's like having a built-in error checker! So, let's take a moment to plug our solution back into the equation and see if it holds true. Remember, math is a journey of discovery, and each step we take brings us closer to understanding the underlying concepts and principles.

Verifying the Solution: Ensuring Accuracy

To make sure our solution x = 6 is correct, we need to substitute it back into the original equation: 2x + 3 = 15. This process, known as verification, is a vital step in solving any equation. It's like double-checking your work to ensure you haven't made any mistakes along the way. Think of it as a safety net, catching any potential errors and solidifying your understanding of the problem. By plugging the value back in, we're essentially asking the question: “Does this value of 'x' truly make the equation true?” If it does, we can be confident in our answer. If it doesn't, we know we need to revisit our steps and identify where we might have gone wrong. So, let's substitute x = 6 into the equation:

2(6) + 3 = 15

Now, let's simplify:

12 + 3 = 15

15 = 15

Fantastic! The equation holds true. This confirms that our solution, x = 6, is indeed correct. Verification not only ensures accuracy but also deepens our understanding of the equation and its solution. It reinforces the concept that the equation represents a balance, and our solution is the value that maintains that balance. This step-by-step substitution process is a powerful tool in mathematics, allowing us to rigorously test our solutions and build a strong foundation for more advanced problem-solving. So, always remember to verify your answers – it's a small step that can make a big difference in your mathematical journey!

Real-World Applications: Where Linear Equations Shine

Okay, we've conquered 2x + 3 = 15, but you might be wondering,