Solving For X: A Step-by-Step Guide

Hey there, math enthusiasts! Let's dive into the equation -4 + 5x - 7 = 10 + 3x - 2x and find the value of x. Solving equations might seem intimidating at first, but trust me, with a few simple steps, you'll be cracking these problems like a pro. This guide will break down the process, making it super easy to understand. We'll go through each step carefully, ensuring you grasp the concepts and can apply them to other similar equations. So, grab your pencils and let's get started on this mathematical adventure! Remember, practice makes perfect, so the more you work through these examples, the better you'll become. By the end of this guide, you'll be well-equipped to handle equations like these with confidence. Let's make math fun and accessible together!

Step 1: Simplify Both Sides of the Equation

Our first step involves simplifying both sides of the equation. This means combining like terms to make the equation less cluttered and easier to manage. Remember, like terms are terms that have the same variable raised to the same power. In our equation, we have constants (numbers without variables) and terms with x. Let's tackle the left side first: -4 + 5x - 7. We can combine the constants -4 and -7. Adding these together, we get -11. The equation now looks like this: 5x - 11. Cool, right? We've simplified one side already!

Now, let's simplify the right side of the equation: 10 + 3x - 2x. Here, we can combine the x terms. We have 3x - 2x, which simplifies to x (or 1x). The right side of the equation becomes 10 + x. Now our equation is looking much cleaner: 5x - 11 = 10 + x. See? Much easier to work with. Simplifying the equation is crucial because it reduces the complexity, allowing us to isolate the variable and solve for x. This simplification process not only makes the equation more manageable but also minimizes the chances of making calculation errors. In the world of algebra, organization and simplification are your best friends. These steps set the stage for efficiently isolating the unknown variable, ensuring we move closer to finding its exact value. Keep in mind that accuracy is key; therefore, double-check each step to avoid any potential mistakes.

Combining Like Terms

Let's zoom in on combining like terms a bit more, as it's a foundational skill for solving equations. Combining like terms involves adding or subtracting terms that have the same variable raised to the same power. For instance, in an expression like 3x + 2x - 5, we can combine 3x and 2x because they both have x to the power of 1. When we combine them, we get 5x. The constant -5 remains as is because it doesn't have a variable attached. Therefore, the simplified expression becomes 5x - 5.

Similarly, if we have an expression like 4y^2 - y^2 + 7, we can combine 4y^2 and -y^2. This results in 3y^2. The constant 7 remains unchanged. The simplified expression would be 3y^2 + 7. Remember, you can only combine terms if they have the same variable and the same exponent. For example, you can't combine 2x and 3x^2. These are different terms, and they must be treated separately. Mastering this skill is incredibly important. It forms the basis for more advanced algebraic manipulations and problem-solving techniques. Practicing with various examples will help you become proficient and build a strong foundation in algebra.

Step 2: Isolate the Variable Terms

Now that we've simplified both sides of the equation, the next step is to isolate the variable terms. This means we want to get all the terms containing x on one side of the equation and the constants on the other side. Let's start by moving the x term from the right side to the left side. Remember that to move a term across the equals sign, we perform the opposite operation. In our equation, 5x - 11 = 10 + x, we have + x on the right side. To move it, we subtract x from both sides of the equation. This gives us 5x - x - 11 = 10 + x - x. Simplifying this, we get 4x - 11 = 10. So far, so good!

Next, we need to move the constant terms. We have -11 on the left side. To get rid of it, we add 11 to both sides of the equation. This gives us 4x - 11 + 11 = 10 + 11. Simplifying this, we get 4x = 21. And there you have it; the variable terms are now isolated on one side, and we're getting closer to solving for x. The goal here is to manipulate the equation to have all x terms on one side and all constant terms on the other side.

Inverse Operations

Let's take a closer look at the concept of inverse operations. Inverse operations are pairs of operations that undo each other. They are fundamental in algebra, enabling us to isolate variables and solve equations. The most common inverse operations are addition and subtraction, multiplication and division. For instance, if you add 5 to a number and then subtract 5, you're back where you started. Similarly, if you multiply a number by 3 and then divide by 3, you also return to the original number. When solving equations, we use inverse operations to