Solving For X: A Step-by-Step Guide

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Hey guys! Let's dive into solving the equation 7 - 5(1 - 2x) = 3(2x + 1). This is a classic algebra problem, and we'll break it down step-by-step so you can totally nail it. Solving for x means we want to isolate x on one side of the equation and find its numerical value. Don't worry if equations make you feel a little uneasy; with practice, it'll become second nature. We'll go through each stage, making sure everything is super clear, so you'll be solving equations like a pro in no time! Ready? Let's do this!

Step 1: Distribute and Simplify

Alright, first things first: we need to get rid of those pesky parentheses. This is where the distributive property comes into play. The distributive property says that you multiply the term outside the parentheses by each term inside the parentheses. So, we'll start by distributing the -5 on the left side and the 3 on the right side of the equation. Remember, a negative times a negative is a positive!

Let's apply this:

  • On the left side, we have -5(1 - 2x). This becomes -5 * 1 and -5 * -2x. That gives us -5 + 10x.
  • On the right side, we have 3(2x + 1). This becomes 3 * 2x and 3 * 1. That gives us 6x + 3.

So, our equation now looks like this: 7 - 5 + 10x = 6x + 3. Next, we simplify the left side by combining the constants (7 and -5), which gives us 2. The equation is now: 2 + 10x = 6x + 3. We’re making progress, guys! It is like peeling back the layers of a math onion, right?

This distribution step is fundamental in algebra. It's like unpacking a box: you're essentially preparing the terms for further manipulation. Make sure you don't miss a term or make any sign errors. Double-check your multiplication. This step is about organizing the terms in a way that allows us to collect like terms and isolate x. Doing it carefully now saves you headaches later. Once you feel comfortable with distribution, you'll find these types of problems much easier.

Remember to stay focused. Math can be tricky sometimes, but taking it slow and being precise will make the whole process much smoother. Always remember the order of operations and make sure you're distributing correctly. You're doing great, and we're just getting started! Keep going, we are almost there. If you're a visual learner, consider writing the distribution process out step by step, so you can track where all the numbers go. Keep practicing, and you will totally get it!

Step 2: Combine Like Terms

In our equation, 2 + 10x = 6x + 3, we need to get all the x terms on one side and all the constant terms on the other. This process is all about collecting like terms. The goal here is to isolate x. We want all the x terms to be together on one side of the equation and all the numbers (constants) on the other side.

  • First, let's move the 6x from the right side to the left side. To do this, we subtract 6x from both sides of the equation. This gives us: 2 + 10x - 6x = 6x - 6x + 3.
  • Simplifying, we get: 2 + 4x = 3.

Now, we move the constant term from the left side to the right side. Subtract 2 from both sides: 2 - 2 + 4x = 3 - 2, which simplifies to 4x = 1. We're getting closer! Combining like terms is a core skill in algebra. It helps us simplify equations into a more manageable form. Think of it like organizing your things at home. You put all the similar items together, right? Like sorting the clothes in your closet or grouping books on your bookshelf. With the x terms and constants now on opposite sides, we are preparing the final step. Always make sure to perform the same operation on both sides of the equation to keep it balanced.

This might seem like a lot of steps, but remember, each step brings you closer to the solution. The key is to be methodical and careful, and you'll always get to the right answer. We’re working towards isolating that x! Keep your eye on the prize. Remember, practice makes perfect. The more you do these kinds of problems, the easier it will become. Don’t get discouraged; everyone starts somewhere. You're building a strong foundation for more complex problems later on. We've got this!

Step 3: Isolate x

Okay, guys, we’re at the final stretch! Our equation is currently 4x = 1. Now, we have to isolate x completely. This means getting x by itself on one side of the equation. To do this, we need to get rid of the 4 that's multiplying x. What's the opposite of multiplication? You got it – division!

  • We divide both sides of the equation by 4. This gives us: (4x) / 4 = 1 / 4.
  • Simplifying, we get: x = 1/4.

And there you have it! We've solved for x. The value of x that satisfies the original equation is 1/4. To double-check our work, we could substitute 1/4 back into the original equation and see if it holds true, but for now, we know the solution. Isolating x is the ultimate goal in solving these types of equations. It's like finding the hidden treasure at the end of a long journey. You've worked through the equation step by step, and now you have the answer. Take a moment to celebrate this success. You did it!

Division is the final step to unlock the value of x. Remember that whatever you do on one side of the equation, you must always do on the other side. This keeps the equation balanced. Keep practicing, and you'll become more and more confident in your ability to solve these kinds of problems. This is an essential skill in mathematics and opens the door to more advanced topics. Knowing how to isolate a variable is key in pretty much all fields of math and science, so this is a win for you.

Step 4: Verification (Optional)

Although we’ve solved for x, it’s always a good idea to check your answer. This step isn't mandatory, but it helps ensure that you got the right answer, right? It's like a final test run to make sure everything works perfectly. We're going to substitute our solution, x = 1/4, back into the original equation and see if the equation holds true. This is called verification. If both sides of the equation are equal after the substitution, then our solution is correct!

Let’s do this:

  • Original equation: 7 - 5(1 - 2x) = 3(2x + 1)
  • Substitute x = 1/4: 7 - 5(1 - 2*(1/4)) = 3(2*(1/4) + 1)

Now, let's simplify step by step:

  • 7 - 5(1 - 1/2) = 3(1/2 + 1)
  • 7 - 5(1/2) = 3(3/2)
  • 7 - 5/2 = 9/2
  • 14/2 - 5/2 = 9/2
  • 9/2 = 9/2

Since both sides of the equation are equal, our solution x = 1/4 is correct! Verifying your answer is a great habit to develop. It not only confirms that your solution is correct but also helps you become more familiar with the equation and the steps you took to solve it. It's like proofreading your work. This is the last step that will give you absolute confidence in your solution, and now you know you're right. So pat yourself on the back, guys – you have successfully solved and verified the equation. It's all about being precise and methodical, and you'll develop a knack for these problems. This skill will pay off big time in the long run.

Conclusion

Alright, awesome job, everyone! We've successfully solved for x in the equation 7 - 5(1 - 2x) = 3(2x + 1). We went through the steps of distributing, combining like terms, and isolating x, and we even verified our answer! You now have a solid understanding of how to tackle these kinds of algebraic equations. Remember, practice is key. The more you work on these problems, the more comfortable and confident you'll become. So, keep at it, and you'll be acing these equations in no time! Keep practicing, and you'll be solving all sorts of math problems like a champ. You're all awesome!