Solving Equations: Step-by-Step Guide

Hey everyone, let's dive into solving the equation: -10x - 10(-x + 1) = 5(5x - 1) - 10. This might look a little intimidating at first, but trust me, with a few simple steps, we can break it down and find the value of x. This guide will walk you through each step, making sure you understand the process completely. We'll cover the fundamentals of algebra, making sure everyone, from beginners to those brushing up on their skills, can follow along. Our focus will be on clarity and ease, avoiding jargon and focusing on the core concepts. Get ready to simplify, solve, and conquer this equation!

Step 1: Distribute and Simplify the Equation

Alright guys, the first thing we need to do is get rid of those parentheses by distributing the numbers outside them. Remember, distribution means multiplying the number outside the parentheses by each term inside. Let’s tackle the left side of the equation first: -10x - 10(-x + 1). We have two parts to deal with here. We'll start by distributing -10 across the terms inside the first set of parentheses. That gives us: -10 * -x and -10 * 1.

So, -10 * -x becomes +10x (because a negative times a negative is a positive), and -10 * 1 is -10. Putting it all together, the left side of the equation simplifies to -10x + 10x - 10. Notice how the -10x and +10x cancel each other out! This leaves us with just -10 on the left side. Okay, on to the right side of the equation: 5(5x - 1) - 10. Here, we distribute the 5 across the terms inside the parentheses: 5 * 5x and 5 * -1. This simplifies to 25x - 5. Don't forget the -10 that was already there! So, the right side becomes 25x - 5 - 10. Combining the constants, -5 - 10 gives us -15. Therefore, the right side simplifies to 25x - 15. Now our equation looks much cleaner, right? We've gone from -10x - 10(-x + 1) = 5(5x - 1) - 10 to -10 = 25x - 15. This is the foundation of solving the equation, making the next steps much easier to handle. Always remember to check your work at this stage; a small mistake can throw off the entire solution. Double-checking ensures accuracy and builds your confidence as you solve more complex problems.

Detailed Breakdown of Distribution

Let’s zoom in on that distribution step for a second. It's super important, and sometimes, it can be a little tricky. Take the term -10(-x + 1). We're essentially saying we have -10 groups of (-x + 1). The distributive property tells us that we multiply each term inside the parentheses by the number outside. So, we do:

• -10 * -x = 10x (Remember, a negative times a negative equals a positive).
• -10 * 1 = -10

So, -10(-x + 1) expands to 10x - 10. Similarly, for 5(5x - 1), we do:

• 5 * 5x = 25x
• 5 * -1 = -5

This gives us 25x - 5. This detailed breakdown ensures you understand exactly how those parentheses disappear. Mastering distribution is crucial for success in algebra, as it’s a fundamental skill used in almost every type of equation. It’s also good practice to write down each step, especially when you're starting. This helps you track your work and spot any potential mistakes. With practice, you'll become a pro at this, and it will become second nature.

Step 2: Isolate the Variable Term

Now that we've simplified, our equation is -10 = 25x - 15. The next step is to isolate the term with the variable, which in this case is 25x. To do this, we need to get rid of that -15 that's hanging out on the right side. We achieve this by performing the opposite operation. Since -15 is being subtracted, we'll add 15 to both sides of the equation. Why both sides? Because in algebra, to keep the equation balanced, whatever you do to one side, you must do to the other. Adding 15 to both sides gives us: -10 + 15 = 25x - 15 + 15.

On the left side, -10 + 15 equals 5. On the right side, -15 + 15 cancels out, leaving us with just 25x. Our equation now looks like this: 5 = 25x. We're getting closer! The goal is always to get the variable (x in this case) all by itself. This process, often referred to as 'isolating the variable', is a cornerstone of equation solving. It's important to understand why we're doing each step. The idea is to undo the operations that are applied to the variable, working backward to get it alone. Remember, the equals sign (=) is like a balance scale. To keep it balanced, you have to treat both sides of the equation the same way. This principle applies to all algebraic manipulations, whether you're adding, subtracting, multiplying, or dividing. Make sure you don't skip steps, especially when you're starting out. Each step helps you build your reasoning skills, making you more confident in solving increasingly complex equations.

Understanding Inverse Operations

Let's take a closer look at inverse operations. They are the heart of isolating the variable. Think of them as the 'undo buttons' in algebra. Addition and subtraction are inverse operations. Multiplication and division are also inverse operations. When we had -15 on the right side, we used addition (+15) to cancel it out. Similarly, if we had +15, we would subtract 15. It's all about finding the operation that 'undoes' what's already there. Here are a few examples to clarify things:

• If you have x + 5 = 10, you subtract 5 from both sides to isolate x.
• If you have x - 3 = 7, you add 3 to both sides.
• If you have 2x = 8, you divide both sides by 2.
• If you have x/4 = 6, you multiply both sides by 4.

Knowing and understanding inverse operations is the key to solving equations efficiently. It’s also a good practice to check your work by substituting the solution back into the original equation, a process that helps confirm the accuracy of your answer. Understanding these concepts will make solving more complex equations feel less daunting and more achievable.

Step 3: Solve for the Variable

We're in the home stretch, guys! Our equation currently stands at 5 = 25x. To solve for x, we need to get x all by itself. Right now, x is being multiplied by 25. To undo that multiplication, we'll do the opposite, which is division. We’ll divide both sides of the equation by 25. This gives us: 5 / 25 = 25x / 25.

On the left side, 5 divided by 25 simplifies to 1/5 (or 0.2 if you prefer decimals). On the right side, 25x divided by 25 leaves us with just x. So, our solution is x = 1/5 (or x = 0.2). Congratulations, you’ve solved the equation! It may seem like a lot of steps, but it’s all about the order. Breaking down the problem into smaller, manageable chunks makes it way easier to solve. Always remember that the goal is to isolate the variable. This will take some practice, but you'll get the hang of it.

Checking Your Solution

Always, and I mean always, check your solution to make sure it’s correct. This is the most crucial part of ensuring your answer is accurate. We'll substitute the value of x (1/5) back into the original equation: -10x - 10(-x + 1) = 5(5x - 1) - 10. Let's start by replacing every x with 1/5:

• -10 * (1/5) - 10(-(1/5) + 1) = 5(5 * (1/5) - 1) - 10

Now, let's simplify each part step by step:

• -10 * (1/5) = -2
• -(1/5) + 1 = 4/5
• 10 * (4/5) = 8
• 5 * (1/5) = 1
• 5 * (1 - 1) = 5 * 0 = 0

So, our equation becomes:

• -2 - 8 = 0 - 10
• -10 = -10

The equation holds true, which means our solution, x = 1/5, is correct! Checking your solution not only confirms your work but also helps build your confidence. If the two sides of the equation don’t match when you substitute the value of x, you know you've made a mistake somewhere, and you can go back and find it. It's all part of the learning process. It is important to perform this step to build good habits and to increase accuracy.

Step 4: Final Answer and Conclusion

So, after all those steps, the final answer to the equation -10x - 10(-x + 1) = 5(5x - 1) - 10 is x = 1/5 (or 0.2). High five to all of you who made it this far!

We started with a complex-looking equation and systematically worked through it, using the principles of distribution, isolating variables, and inverse operations. We even checked our answer to ensure it's correct. Solving equations might seem daunting, but by breaking it down into smaller, more manageable steps, and using the right tools (like distribution and inverse operations), anyone can master it. Remember to always double-check your work and practice regularly. The more you practice, the easier it becomes. Keep up the great work, and you'll be solving equations like a pro in no time! Keep practicing, and you'll become a pro in no time! Keep learning, keep practicing, and keep that mathematical spirit alive. Now go forth and conquer more equations!