Solving For X: -4 + 5x - 7 = 10 + 3x - 2x

by Alex Johnson 42 views

Let's dive into the world of algebra and tackle this equation together! If you've ever felt a little intimidated by equations, don't worry. We're going to break it down step-by-step, so it's super easy to follow. The equation we're working with today is -4 + 5x - 7 = 10 + 3x - 2x. Our goal? To find the value of x that makes this equation true. Solving for x in linear equations like this involves using algebraic manipulation to isolate x on one side of the equals sign. This means simplifying the equation by combining like terms, and then performing the same operations on both sides to maintain the balance of the equation.

Understanding the Equation

Before we start crunching numbers, let's take a good look at what we have. The equation -4 + 5x - 7 = 10 + 3x - 2x might look a bit complex at first glance, but it's really just a balancing act. We have two sides, separated by the equals sign (=). Think of it like a scale – both sides need to weigh the same for the equation to be true. On the left side, we have a mix of constant numbers (-4 and -7) and a term with x (5x). On the right side, we also have a constant (10) and terms with x (3x and -2x). Our mission is to simplify each side and then isolate x so we can find its value. When dealing with algebraic equations, it is important to remember the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). In our case, we will focus on simplifying by combining like terms, which primarily involves addition and subtraction.

Step-by-Step Solution

Okay, let's get to work! We'll break this down into manageable steps to make it nice and clear.

1. Simplify Each Side

The first thing we want to do is tidy up each side of the equation. This means combining any like terms – constants with constants, and x terms with x terms. On the left side, we have -4 and -7, which are both constants. Let's combine them: -4 - 7 = -11. So, the left side simplifies to 5x - 11. Now, let's look at the right side. We have 3x and -2x, which are both x terms. Combining them gives us 3x - 2x = x. So, the right side simplifies to 10 + x. Our equation now looks much cleaner: 5x - 11 = 10 + x. This step is crucial because it condenses the equation into a more manageable form. By combining like terms, we reduce the number of individual elements in the equation, making it easier to see the relationship between the variables and constants. It's like decluttering a messy room – once everything is organized, it's much easier to find what you're looking for.

2. Get the x Terms on One Side

Now, we want to get all the x terms together on one side of the equation. It doesn't matter which side we choose, but let's aim for the side that will give us a positive coefficient for x (just to make things a little easier). In this case, we have 5x on the left and x on the right. To get rid of the x on the right side, we'll subtract x from both sides. Remember, whatever we do to one side, we must do to the other to keep the equation balanced! So, we have: 5x - 11 - x = 10 + x - x. This simplifies to 4x - 11 = 10. By subtracting x from both sides, we've successfully moved all the x terms to the left side of the equation. This is a key step in isolating x and solving for its value. Keeping the equation balanced is like making sure both sides of a seesaw are level – if you add or take away weight from one side, you need to do the same on the other side to maintain equilibrium.

3. Isolate the x Term

We're getting closer! Now we need to isolate the x term completely. This means getting rid of the -11 on the left side. To do that, we'll add 11 to both sides (again, keeping things balanced): 4x - 11 + 11 = 10 + 11. This simplifies to 4x = 21. Now we have the x term all by itself on the left side, which is exactly what we want. Adding 11 to both sides was the key to isolating the x term. This step is like peeling away the layers of an onion – we're gradually stripping away everything that's not x until we have x all by itself. Think of it as moving all the other puzzle pieces away so you can focus on the one you need.

4. Solve for x

Finally, the moment we've been waiting for! To solve for x, we need to get rid of the 4 that's multiplying it. We do this by dividing both sides by 4: 4x / 4 = 21 / 4. This simplifies to x = 21/4. And there you have it! We've found the value of x that satisfies the equation. Dividing both sides by 4 was the final step in unwrapping the mystery of x. This is like the last piece of the puzzle clicking into place – you've done all the hard work, and now you have the solution. Always double-check your answer by plugging it back into the original equation to ensure it holds true. This can help you catch any errors made during the solving process.

The Answer

So, the solution for x in the equation -4 + 5x - 7 = 10 + 3x - 2x is x = 21/4. Looking back at the options, the correct answer is C. x = 21/4. We did it! We successfully navigated the equation and found the value of x. This is a fantastic example of how breaking down a problem into smaller, manageable steps can make even complex-looking equations solvable. Remember, solving for x is like following a treasure map – each step leads you closer to the prize. And with a little practice, you'll become a pro at finding that treasure!

Key Concepts Recap

Let's quickly recap the key concepts we used to solve this equation. These are fundamental principles in algebra, and mastering them will help you tackle all sorts of mathematical challenges.

  • Combining Like Terms: This is the foundation of simplifying equations. We combined constants with constants and x terms with x terms to tidy up each side of the equation. This is like organizing your workspace before starting a project – it makes everything clearer and more manageable.
  • Maintaining Balance: Remember the seesaw analogy? Whatever operation we perform on one side of the equation, we must do the same on the other side to keep it balanced. This is crucial for ensuring that our solution remains valid. Think of it as the golden rule of algebra – treat both sides equally!
  • Inverse Operations: To isolate x, we used inverse operations. Subtraction undoes addition, and division undoes multiplication. This is like using the right tool for the job – you wouldn't use a hammer to unscrew a bolt, and you wouldn't use addition to undo multiplication.
  • Step-by-Step Approach: We broke the problem down into smaller, more manageable steps. This made the whole process less daunting and easier to follow. This is a valuable problem-solving skill that can be applied in many areas of life – tackling a big task one step at a time.

Practice Makes Perfect

Like any skill, solving equations gets easier with practice. The more you work with algebraic equations, the more comfortable you'll become with the process. Don't be afraid to make mistakes – they're a natural part of learning. Each mistake is an opportunity to understand the concepts more deeply and refine your approach. Try working through similar equations, experimenting with different techniques, and challenging yourself with progressively more complex problems. There are tons of resources available online and in textbooks to help you practice. You can also try creating your own equations and solving them, or working with a friend or tutor to tackle problems together. Remember, the key is to keep practicing and keep learning! With consistent effort, you'll become a confident and skilled equation solver.

Conclusion

We've successfully solved for x in the equation -4 + 5x - 7 = 10 + 3x - 2x! By following a step-by-step approach, combining like terms, and using inverse operations, we found that x = 21/4. This journey through algebra highlights the power of breaking down complex problems into manageable steps. Remember, mathematics is not just about finding the right answer, it's about developing problem-solving skills that can be applied in all areas of life. Keep practicing, keep exploring, and keep challenging yourself – you've got this! For further learning and practice, you might find helpful resources on websites like Khan Academy's Algebra Section. Happy solving!