Solving For X: A Step-by-Step Guide

Have you ever stared at an equation and felt completely lost? Don't worry, it happens to the best of us! Solving for 'x' is a fundamental skill in algebra, and it's like learning a secret code to unlock mathematical problems. In this guide, we'll break down the equation 5x + 8 + x = 32 into simple, easy-to-follow steps. Think of it as a puzzle where we need to isolate 'x' on one side of the equation to find its value. So, grab your pencil and paper, and let's embark on this mathematical adventure together! We will cover every aspect, from understanding the basics to applying the correct operations, ensuring you grasp the concept thoroughly. Let’s dive in and make algebra less intimidating and more approachable.

Understanding the Basics of Algebraic Equations

Before we jump into the specifics of solving 5x + 8 + x = 32, let's establish a solid foundation by understanding the basics of algebraic equations. An algebraic equation is a mathematical statement that shows the equality of two expressions. It contains variables (usually represented by letters like 'x', 'y', or 'z'), constants (numbers), and mathematical operations (+, -, ×, ÷). The goal of solving an equation is to find the value(s) of the variable(s) that make the equation true. Think of an equation as a balanced scale; both sides must remain equal to maintain the balance. Any operation you perform on one side must also be performed on the other side to keep the equation balanced. For instance, if you add a number to the left side, you must add the same number to the right side. This principle is crucial for solving equations correctly. Understanding the properties of equality, such as the addition, subtraction, multiplication, and division properties, is essential. These properties allow us to manipulate equations while maintaining their balance. Grasping these fundamentals will not only help you solve this particular equation but also empower you to tackle more complex algebraic problems in the future. So, let’s keep these basics in mind as we move forward.

Step 1: Combining Like Terms

The first step in solving 5x + 8 + x = 32 is to combine like terms. Like terms are terms that have the same variable raised to the same power. In this equation, we have two terms with 'x': 5x and x. We can combine these terms by adding their coefficients (the numbers in front of the variable). So, 5x + x becomes 6x. Remember, when a variable stands alone, it's understood to have a coefficient of 1. Therefore, x is the same as 1x. Now, our equation looks simpler: 6x + 8 = 32. Combining like terms is a crucial step because it simplifies the equation, making it easier to isolate the variable. This process reduces the number of terms we need to deal with, which in turn reduces the complexity of the problem. It’s like organizing your workspace before starting a task; a clear equation is much easier to solve than a cluttered one. By combining like terms, we’ve taken a significant step towards finding the value of 'x'. This method is universally applicable in algebra, making it a valuable tool in your problem-solving arsenal. So, always look for like terms first, and combine them to simplify your equation.

Step 2: Isolating the Variable Term

Now that we've combined like terms and have the equation 6x + 8 = 32, our next goal is to isolate the variable term. This means we want to get the term with 'x' (which is 6x) alone on one side of the equation. To do this, we need to eliminate the constant term (+8) on the left side. We can achieve this by using the subtraction property of equality. This property states that we can subtract the same number from both sides of the equation without changing the equality. So, we subtract 8 from both sides of the equation: 6x + 8 - 8 = 32 - 8. This simplifies to 6x = 24. Notice how the +8 and -8 on the left side cancel each other out, leaving us with just the variable term. Isolating the variable term is a critical step because it brings us closer to finding the value of 'x'. By eliminating the constant term, we create a simpler equation that involves only the variable term and a constant on the other side. This step sets us up perfectly for the final step, where we'll solve for 'x' directly. So, remember, the key to isolating the variable term is to use inverse operations to cancel out any constants on the same side of the equation.

Step 3: Solving for x

We've reached the final step in solving for 'x'! Our equation is now 6x = 24. To solve for 'x', we need to isolate 'x' completely. Currently, 'x' is being multiplied by 6. To undo this multiplication, we use the inverse operation, which is division. The division property of equality states that we can divide both sides of the equation by the same non-zero number without changing the equality. So, we divide both sides of the equation by 6: 6x / 6 = 24 / 6. This simplifies to x = 4. And there you have it! We've successfully solved for 'x'. The value of 'x' that makes the original equation true is 4. This final step demonstrates the power of using inverse operations to isolate the variable and find its value. It’s the culmination of all the previous steps, where we simplified the equation and isolated the variable term. Solving for 'x' is the ultimate goal, and by dividing both sides by the coefficient of 'x', we achieve that goal. So, remember, when solving for a variable, always look for the operation being performed on it and use the inverse operation to isolate it.

Checking Your Solution

Before we celebrate our victory, it's always a good idea to check our solution. This ensures that we haven't made any mistakes along the way. To check our solution, we substitute the value we found for 'x' (which is 4) back into the original equation: 5x + 8 + x = 32. Substituting x = 4, we get 5(4) + 8 + 4 = 32. Now, we simplify the left side of the equation: 20 + 8 + 4 = 32. Adding the numbers, we get 32 = 32. Since both sides of the equation are equal, our solution is correct! Checking your solution is a crucial step in problem-solving. It gives you confidence in your answer and helps you identify any errors you might have made. This process reinforces your understanding of the equation and the steps you took to solve it. It’s like proofreading your work before submitting it; it ensures accuracy and completeness. So, always take the time to check your solution by substituting it back into the original equation. This practice will help you become a more confident and accurate problem solver.

Conclusion

Congratulations! You've successfully solved for 'x' in the equation 5x + 8 + x = 32. We've journeyed through the steps of combining like terms, isolating the variable term, and finally, solving for 'x'. Remember, the key to solving algebraic equations is to follow a systematic approach: simplify, isolate, and solve. And don't forget to check your solution! Solving for 'x' is a fundamental skill in algebra, and mastering it opens the door to more advanced mathematical concepts. Keep practicing, and you'll become more confident and proficient in solving equations. Math can be challenging, but with patience and the right approach, you can conquer any problem. Embrace the challenge, enjoy the process, and celebrate your successes. Now that you've tackled this equation, you're well-equipped to face many more. Happy solving!

For further learning and practice on algebraic equations, visit Khan Academy's Algebra Section.