Solving Equations: A Step-by-Step Guide

Hey there, math enthusiasts! Today, we're diving into the world of equations, specifically tackling the problem 7(x-2) = 4(-5). Don't worry if equations seem daunting at first; with a clear understanding of the steps, you'll be solving them like a pro in no time! We'll break down the process into easy-to-follow steps, ensuring you grasp the concepts and build your problem-solving skills. Understanding how to solve these kinds of equations is super important; it forms the foundation for more complex mathematical concepts you'll encounter later on. We'll start with the basics, making sure we cover everything you need to know. So, grab your pens and paper, and let's get started! The goal is to isolate the variable, which in this case is 'x,' on one side of the equation. This involves a series of algebraic manipulations that we will explain in detail. Each step is designed to simplify the equation, bringing us closer to the solution. The principles of equality are key: whatever you do to one side of the equation, you must do to the other to maintain balance. This ensures that the equation remains valid throughout the solution process. Let's get our hands dirty and start solving it! Learning to solve this type of equation is the first step in the journey of mathematics. In this section, we'll explain how to find the answer step by step. This is a basic algebraic problem. Solving an equation means finding the value of the unknown variable, in this case 'x', that makes the equation true. The equation is like a balanced scale; to keep it balanced, any operation you perform on one side must also be done on the other. That's the gist of it! We'll start by simplifying, then isolate the variable, and finally, we'll find the value of x. By following this method, it will be easy to solve the problem and also any equation problem. Let's see how!

Step-by-Step Solution

Alright, let's break down the equation 7(x-2) = 4(-5) step-by-step to find the value of 'x'. We'll make sure to cover every aspect so that you can solve the same type of equations in the future. We'll show you exactly how to do it! Keep your eyes on the steps, and then you can try to solve more complex problems with ease. The first step involves simplifying both sides of the equation. We need to eliminate the parentheses on the left side and multiply the numbers on the right side. This will make our equation easier to handle. Then, we need to focus on isolating the variable 'x' to find its value. So, let’s go!

Step 1: Distribute the 7

First, we need to distribute the 7 across the terms inside the parentheses. This means multiplying 7 by both 'x' and '-2'. Remember, distribution is key to simplifying this part of the equation. This step is crucial for removing the parentheses and making the equation easier to solve. When we distribute, we get 7 * x - 7 * 2. It's like saying you have 7 groups, and each group has 'x' and '-2'. So, you multiply each term inside the parentheses by 7. Keep in mind that we're keeping the basic rules of algebra. This step is about expanding the left side of the equation. It's a fundamental operation in algebra and it helps us to solve the problem. Let’s do it now. So, the left side of the equation 7(x-2) becomes 7x - 14. This distribution transforms the equation, making it more manageable for the next steps. Now, the equation looks like this: 7x - 14 = 4(-5). Nice and simple, right?

Step 2: Simplify the Right Side

Now, let's simplify the right side of the equation. We have 4(-5), which is simply a multiplication problem. Multiplying 4 by -5, we get -20. This simplifies the equation further. Keep in mind that multiplying a positive number by a negative number results in a negative number. This is one of the important rules to keep in mind! Now our equation will look like this: 7x - 14 = -20. We're gradually getting closer to solving for 'x'. See how we're making the equation cleaner and easier to solve? The basic arithmetic operations are essential. In this step, we've brought the right side of the equation to its simplest form. This makes it easier to work with in the next steps. So, let's move on to the next one!

Step 3: Isolate the Variable Term

Next, we need to isolate the term with the variable, which is 7x. To do this, we'll add 14 to both sides of the equation. Remember, whatever we do to one side, we must do to the other to keep the equation balanced. Adding 14 cancels out the -14 on the left side, leaving us with just 7x. This principle is fundamental in algebra, as it maintains the equality of the equation. So, adding 14 to both sides, we get: 7x - 14 + 14 = -20 + 14. On the left side, -14 and +14 cancel each other out, leaving us with 7x. On the right side, -20 + 14 equals -6. Therefore, the equation becomes 7x = -6. This isolates the variable term, bringing us a step closer to solving for 'x'.

Step 4: Solve for x

Now it's time to solve for 'x'. We have 7x = -6. To isolate 'x', we need to divide both sides of the equation by 7. This is the final step to find the value of x. Dividing both sides by 7, we get: (7x) / 7 = -6 / 7. The 7s on the left side cancel each other out, leaving us with 'x'. On the right side, -6 divided by 7 is -6/7. So, the solution is x = -6/7. And that’s our answer, guys! We have successfully solved the equation 7(x-2) = 4(-5).

Conclusion: The Final Answer

Great job, everyone! We've successfully navigated through the equation 7(x-2) = 4(-5), step by step, and found the solution. The value of x is -6/7. Always remember the steps: distribute, simplify, isolate, and solve. By following these, you can tackle any equation. Practicing these steps will help you master algebraic equations. Mathematics may seem difficult at first, but with practice, you can get there. You're now equipped with the skills to solve similar problems. Keep practicing, and you'll find that solving equations becomes easier and more intuitive. Keep up the great work, and happy solving!