Solving The Equation 6(x-8) + X = 7(x-8) + 8 A Step-by-Step Guide

by James Vasile 66 views

Let's dive into solving the equation 6(x - 8) + x = 7(x - 8) + 8 together! Equations like this might seem a bit daunting at first glance, but don't worry, we'll break it down step by step. Our main goal here is to isolate the variable 'x' on one side of the equation. Think of it like a puzzle – we're just rearranging the pieces until we find the solution. First, we need to deal with those parentheses. Remember the distributive property? We're going to use that to multiply the numbers outside the parentheses by each term inside. This will help us simplify the equation and make it easier to work with. So, let's get started and unravel this mathematical mystery! Solving equations is a fundamental skill in algebra, and mastering it will open doors to more advanced concepts. By understanding the underlying principles and applying them systematically, you'll find that even complex equations become manageable. Remember, practice is key! The more you solve equations, the more comfortable and confident you'll become. And don't be afraid to make mistakes – they're part of the learning process. Just keep trying, and you'll get there. Now, let's get back to our equation and see how we can solve it.

Step-by-Step Solution

1. Apply the Distributive Property

The distributive property is our first key tool. We'll apply it to both sides of the equation. This involves multiplying the number outside the parentheses by each term inside. For the left side, we have 6(x - 8). Multiplying 6 by x gives us 6x, and multiplying 6 by -8 gives us -48. So, 6(x - 8) becomes 6x - 48. Now, let's look at the right side of the equation: 7(x - 8). Similarly, we multiply 7 by x, which gives us 7x, and 7 by -8, which gives us -56. So, 7(x - 8) becomes 7x - 56. After applying the distributive property, our equation now looks like this: 6x - 48 + x = 7x - 56 + 8. See? We've already made progress! The parentheses are gone, and the equation is starting to look simpler. This step is crucial because it allows us to combine like terms, which will further simplify the equation. Remember, the distributive property is a fundamental concept in algebra, and it's used extensively in solving equations and simplifying expressions. So, make sure you're comfortable with it. If you're not, try practicing with a few more examples. The more you practice, the better you'll become at it. And once you've mastered the distributive property, you'll be well on your way to solving more complex equations. Now, let's move on to the next step and see how we can further simplify our equation.

2. Combine Like Terms

Now, let's simplify each side by combining like terms. On the left side of the equation, we have 6x and x. These are like terms because they both contain the variable 'x'. When we combine them, we simply add their coefficients (the numbers in front of the variable). In this case, the coefficient of 6x is 6, and the coefficient of x is 1 (since x is the same as 1x). So, 6x + x becomes 7x. The left side of the equation now looks like 7x - 48. On the right side, we have -56 and 8. These are like terms because they are both constants (numbers without variables). When we combine them, we simply add them together. -56 + 8 equals -48. So, the right side of the equation now looks like 7x - 48. After combining like terms, our equation is now: 7x - 48 = 7x - 48. Wow, things are getting interesting! Notice anything special about this equation? Both sides are exactly the same! This is a clue about the type of solution we're going to get. But before we jump to conclusions, let's continue with our steps to solve for 'x'. Combining like terms is another essential skill in algebra. It helps us to simplify equations and make them easier to solve. Remember, you can only combine terms that have the same variable and exponent (like 6x and x) or that are constants (like -56 and 8). Now, let's move on to the next step and see what we can learn from this unique equation.

3. Isolate the Variable

Our next goal is to isolate the variable 'x' on one side of the equation. To do this, we need to get all the terms with 'x' on one side and all the constant terms on the other side. Let's start by subtracting 7x from both sides of the equation. This will eliminate the 'x' term from the right side. When we subtract 7x from the left side (7x - 48), we get 7x - 7x - 48, which simplifies to -48. When we subtract 7x from the right side (7x - 48), we get 7x - 7x - 48, which simplifies to -48. So, after subtracting 7x from both sides, our equation becomes: -48 = -48. Hmm, this is interesting. The 'x' variable has disappeared completely! We're left with a statement that is always true: -48 is equal to -48. This tells us something important about the solution to the original equation. When we end up with a true statement like this, it means that the equation has infinitely many solutions. In other words, any value of 'x' will make the equation true. This is a special case that can sometimes occur when solving equations. It's important to recognize it when it happens, so you can correctly interpret the solution. Isolating the variable is a fundamental technique in solving equations. It involves using inverse operations (like addition and subtraction, or multiplication and division) to get the variable by itself on one side of the equation. Now, let's state our conclusion based on what we've found.

4. Interpret the Solution

So, what does -48 = -48 mean? It means that the original equation, 6(x - 8) + x = 7(x - 8) + 8, is true for any value of x. This is because, as we simplified the equation, the variable 'x' completely disappeared, leaving us with a true statement. When this happens, we say that the equation has infinitely many solutions. Think about it this way: no matter what number you plug in for 'x', the equation will always balance out. This is a different kind of solution than what we usually find when solving equations. Usually, we get a specific value for 'x' that makes the equation true. But in this case, any value works! This type of equation is called an identity. An identity is an equation that is true for all values of the variable. Recognizing an identity is a valuable skill in algebra. It helps us understand the nature of the equation and its solutions. When you encounter an equation that simplifies to a true statement with no variables, you know you're dealing with an identity. And that means you have infinitely many solutions. So, to summarize, the solution to the equation 6(x - 8) + x = 7(x - 8) + 8 is: All real numbers. This means that any number you can think of will satisfy the equation. Congratulations! You've successfully solved this equation and learned about a special case in equation solving. Now, let's put it all together in a final answer.

Final Answer

The solution to the equation 6(x - 8) + x = 7(x - 8) + 8 is:

x = All real numbers

This indicates that the equation is an identity, meaning it holds true for any value of 'x'. We've walked through the steps, from distributing and combining like terms to recognizing the special case of infinitely many solutions. Solving equations is a journey, and with each problem, we learn something new. Remember, practice makes perfect! The more you work with equations, the more comfortable you'll become with the different techniques and types of solutions. Don't be afraid to tackle challenging problems, and always double-check your work to ensure accuracy. Now that you've mastered this equation, you're ready to take on even more complex algebraic challenges. Keep exploring, keep learning, and keep solving!