Solving 5x - 10 = 0: A Step-by-Step Guide

Hey guys! Today, we're going to tackle a common algebra problem: solving for a variable in a two-step equation. Specifically, we'll break down how to solve the equation 5x - 10 = 0 step-by-step. Don't worry, it's easier than it looks! We'll use the properties of equality to isolate 'x' and find our solution. Think of it like a puzzle – we just need to carefully undo the operations to reveal the answer. By understanding the underlying principles and the order of operations, we can confidently solve a wide range of algebraic equations. So, let's dive in and learn how to conquer this equation together!

Understanding Two-Step Equations

First, let's make sure we're all on the same page about what a two-step equation actually is. These are algebraic equations that, you guessed it, require two steps to solve. They typically involve a variable multiplied by a coefficient (the number in front of the variable) and then either added or subtracted by a constant. Our equation, 5x - 10 = 0, perfectly fits this description. The goal in solving any equation is to isolate the variable – in this case, 'x' – on one side of the equals sign. We achieve this by using inverse operations and the properties of equality. These properties of equality are the fundamental rules that allow us to manipulate equations without changing their solutions. Mastering these principles is crucial for success in algebra and beyond. Think of them as the secret tools in your mathematical toolkit, ready to be used whenever you encounter an equation that needs solving. Remember, math is like learning a new language, and each concept builds upon the previous one. So, a solid understanding of two-step equations lays the groundwork for tackling more complex problems later on.

Step 1: Using the Addition Property of Equality

The first step in solving 5x - 10 = 0 is to isolate the term with the variable (5x). To do this, we need to get rid of the -10. This is where the addition property of equality comes into play. This property states that you can add the same value to both sides of an equation without changing the equality. In our case, we'll add 10 to both sides: 5x - 10 + 10 = 0 + 10. On the left side, the -10 and +10 cancel each other out, leaving us with 5x. On the right side, 0 + 10 equals 10. So, our equation now becomes 5x = 10. See? We're already one step closer to solving for 'x'! This first step highlights the power of inverse operations. We used addition to undo the subtraction, effectively moving the constant term to the other side of the equation. This strategic manipulation is key to simplifying equations and isolating the variable. Always remember to perform the same operation on both sides to maintain the balance and ensure the equation remains true.

Step 2: Applying the Division Property of Equality

Now that we have 5x = 10, we need to isolate 'x' completely. 'x' is currently being multiplied by 5, so to undo this, we'll use the division property of equality. This property states that you can divide both sides of an equation by the same non-zero value without changing the equality. In this case, we'll divide both sides by 5: (5x) / 5 = 10 / 5. On the left side, the 5s cancel each other out, leaving us with just 'x'. On the right side, 10 divided by 5 is 2. Therefore, our solution is x = 2. We've successfully solved for 'x'! This second step demonstrates the importance of understanding the relationship between multiplication and division. They are inverse operations, and we used division to undo the multiplication and finally isolate our variable. The division property of equality is a powerful tool in our algebraic arsenal, allowing us to simplify equations and arrive at the solution. Always double-check that you're dividing by the coefficient of the variable to effectively isolate it.

Checking Your Solution

It's always a good idea to check your solution to make sure it's correct. To do this, we'll substitute our solution (x = 2) back into the original equation: 5x - 10 = 0. Substituting 'x' with 2, we get 5(2) - 10 = 0. Simplifying, we have 10 - 10 = 0, which is true. This confirms that our solution, x = 2, is correct. Checking your work is a critical step in problem-solving. It helps you catch any errors you might have made along the way and ensures that your answer is accurate. Think of it as a safety net – it provides confidence in your solution and reinforces your understanding of the problem. By substituting the solution back into the original equation, you're essentially reversing the steps you took to solve it, which can further solidify your comprehension of the process.

Why are Properties of Equality Important?

The properties of equality are the bedrock of solving algebraic equations. They provide the justification for each step we take. Without them, we wouldn't be able to manipulate equations in a consistent and reliable way. The addition property allows us to add or subtract the same value from both sides, while the division property allows us to divide both sides by the same non-zero value. These properties ensure that we maintain the balance of the equation, meaning that the two sides remain equal throughout the solving process. Understanding these properties is not just about memorizing rules; it's about grasping the underlying logic of equation solving. When you understand why you're doing something, you're much more likely to remember it and apply it correctly in different situations. The properties of equality are like the grammar of algebra – they provide the structure and rules for writing mathematically correct statements and solutions.

Real-World Applications of Solving Equations

Solving equations isn't just a theoretical exercise; it has numerous real-world applications. From calculating the cost of items on sale to determining the trajectory of a rocket, equations are used in countless fields. In finance, equations are used to calculate interest rates and loan payments. In physics, they're used to model motion and energy. In engineering, they're used to design structures and circuits. The ability to solve equations is a valuable skill that can be applied to a wide range of problems. Consider a scenario where you're trying to budget your monthly expenses. You might use an equation to determine how much you can spend on entertainment after accounting for rent, utilities, and other fixed costs. Or, imagine you're a scientist conducting an experiment. You might use equations to analyze your data and draw conclusions. The applications are truly endless. By mastering the art of equation solving, you're equipping yourself with a powerful tool that can help you navigate the complexities of the world around you.

Conclusion

So, there you have it! We've successfully solved the equation 5x - 10 = 0 by using the addition and division properties of equality. Remember, the key is to isolate the variable by performing inverse operations on both sides of the equation. Don't be afraid to practice and try different problems. The more you practice, the more comfortable and confident you'll become in your equation-solving abilities. And always remember to check your solutions! Solving equations is a fundamental skill in mathematics, and mastering it will open doors to more advanced topics and real-world applications. Keep practicing, keep learning, and you'll become a math whiz in no time! You got this, guys! Remember that understanding the underlying principles like the properties of equality is more important than just memorizing steps. This knowledge will empower you to tackle various mathematical challenges with confidence. Keep exploring, keep asking questions, and never stop learning!