Solving Linear Equations: A Step-by-Step Guide
Alright, guys, let's dive into solving a simple linear equation. Today, we're tackling the equation 2x + 3 - 5 * 2 * x + 4 = 0. Don't worry, it's not as scary as it looks! We'll break it down step by step, making sure everyone can follow along.
Understanding the Basics
Before we jump into the solution, let's quickly recap what a linear equation is. A linear equation is an equation in which the highest power of the variable (in this case, x) is 1. These equations can be represented in the form ax + b = 0, where a and b are constants. Our goal is to find the value of x that makes the equation true.
Now, let's look at our equation again: 2x + 3 - 5 * 2 * x + 4 = 0. The first thing you'll notice is that we have multiple terms involving x and some constant terms. Our strategy will be to simplify the equation by combining like terms. Like terms are terms that have the same variable raised to the same power. In our case, the like terms are the terms with x and the constant terms (numbers without x).
Why is this important? Because combining like terms makes the equation easier to work with. It reduces the number of terms we have to deal with, which simplifies the algebraic manipulations we need to perform. Think of it like organizing your room: putting similar items together makes everything easier to find and manage. Similarly, in math, combining like terms makes the equation more manageable and easier to solve. This is a fundamental principle in algebra, and mastering it will help you tackle more complex equations in the future. Remember, the key is to identify the terms that can be combined and then perform the necessary arithmetic to simplify the equation. So, let's get started!
Step 1: Simplify the Equation
The first step is to simplify the equation by performing the multiplication. We have 5 * 2 * x, which simplifies to 10x. So, our equation becomes:
2x + 3 - 10x + 4 = 0
Now, we need to combine the like terms. We have two terms with x: 2x and -10x. Combining these gives us 2x - 10x = -8x. We also have two constant terms: 3 and 4. Combining these gives us 3 + 4 = 7. So, the simplified equation is:
-8x + 7 = 0
Simplifying equations is like decluttering a room; it makes everything easier to manage. By combining like terms, we reduce the number of individual elements we need to handle, making the subsequent steps smoother and more efficient. In this case, we went from having four separate terms to just two. This not only makes the equation look less intimidating but also sets us up for a straightforward isolation of the variable x. This step is crucial because it transforms a potentially confusing equation into a manageable form, paving the way for a quick and accurate solution. Remember, always look for opportunities to simplify before moving on; it's a fundamental strategy that applies to all sorts of algebraic problems. This approach not only saves time but also minimizes the risk of errors. So, keep an eye out for those like terms and combine away!
Step 2: Isolate the Variable
Our goal is to get x by itself on one side of the equation. To do this, we need to isolate the term with x. We have -8x + 7 = 0. First, we'll subtract 7 from both sides of the equation to get rid of the 7 on the left side:
-8x + 7 - 7 = 0 - 7
This simplifies to:
-8x = -7
Isolating the variable is like carefully extracting a single piece from a complex puzzle; it requires precision and focus. In this case, we want to get x all by itself on one side of the equation, which means removing any other terms that are cluttering its space. We achieve this by performing inverse operations. Since we have -8x + 7 = 0, we first deal with the +7 by subtracting 7 from both sides. This maintains the balance of the equation while moving us closer to our goal. Remember, whatever you do to one side of the equation, you must do to the other. This ensures that the equation remains true and that we're not altering the fundamental relationship between the two sides. The result, -8x = -7, is a significant step forward. We've successfully isolated the term with x, setting the stage for the final step: solving for x itself. This methodical approach is key to mastering algebra and solving even more complex equations with confidence.
Step 3: Solve for x
Now that we have -8x = -7, we need to solve for x. To do this, we'll divide both sides of the equation by -8:
-8x / -8 = -7 / -8
This simplifies to:
x = 7/8
So, the solution to the equation 2x + 3 - 5 * 2 * x + 4 = 0 is x = 7/8.
Solving for x is the grand finale of our algebraic journey, the moment when we finally uncover the hidden value of our variable. After isolating the term with x, we're left with a simple equation where x is multiplied by a coefficient. To undo this multiplication, we perform the inverse operation: division. In our case, we have -8x = -7, so we divide both sides by -8. Remember, the golden rule of algebra: whatever you do to one side, you must do to the other. This ensures that the equation remains balanced and that we arrive at the correct solution. The division yields x = 7/8, which means that the value of x that satisfies the original equation is seven-eighths. This is a precise and accurate answer, and we've arrived at it through a series of logical and methodical steps. This final step is a testament to the power of algebra and our ability to solve for the unknown. With practice and patience, you can master these techniques and confidently tackle any equation that comes your way. Congratulations, you've solved it!
Verification (Optional but Recommended)
To make sure our solution is correct, we can plug x = 7/8 back into the original equation and see if it holds true:
2 * (7/8) + 3 - 5 * 2 * (7/8) + 4 = 0
(14/8) + 3 - (70/8) + 4 = 0
(7/4) + 3 - (35/4) + 4 = 0
(7/4) - (35/4) + 7 = 0
(-28/4) + 7 = 0
-7 + 7 = 0
0 = 0
The equation holds true, so our solution x = 7/8 is correct!
Verifying our solution is like double-checking our work after completing a task; it's an essential step that ensures accuracy and confidence in our results. By plugging the value we found for x back into the original equation, we can confirm whether it satisfies the equation or not. In our case, we substitute x = 7/8 into 2x + 3 - 5 * 2 * x + 4 = 0 and simplify. The process involves performing the arithmetic operations, combining like terms, and ultimately verifying whether the left-hand side of the equation equals the right-hand side. If the equation holds true, it confirms that our solution is correct. If not, it indicates that we need to revisit our steps and identify any potential errors. This verification step not only validates our solution but also reinforces our understanding of the equation and the algebraic manipulations we performed. It's a valuable practice that helps build confidence and accuracy in solving equations. So, always take the time to verify your solutions; it's a small investment that can save you from potential mistakes.
Conclusion
Solving linear equations might seem daunting at first, but by breaking them down into simple steps, anyone can do it! Remember to simplify, isolate the variable, and solve for x. And always verify your solution to make sure you got it right. Keep practicing, and you'll become a pro in no time!
So there you have it! You've successfully solved a linear equation. Remember, practice makes perfect. The more you solve these types of problems, the easier it will become. Keep up the great work, and you'll be a math whiz in no time! Keep up the great work!