Solving For X: A Step-by-Step Guide To 8 + 4 = 2(x - 1)

by Alex Johnson 56 views

Are you ready to dive into the world of algebra and solve for x? This equation, 8 + 4 = 2(x - 1), might look intimidating at first glance, but fear not! We'll break it down step-by-step, making it clear and easy to understand. Whether you're a student tackling homework or just brushing up on your math skills, this guide will walk you through the process.

Understanding the Basics

Before we jump into solving, let's quickly review some fundamental concepts. Algebra is all about using symbols and letters to represent numbers and quantities. In our equation, 'x' is the unknown variable we want to find. The equation itself states that the expression on the left side (8 + 4) is equal to the expression on the right side (2(x - 1)). Our goal is to isolate 'x' on one side of the equation to determine its value.

To do this, we'll use the principles of algebraic manipulation. This involves performing the same operations on both sides of the equation to maintain balance. Think of it like a scale: if you add or subtract something on one side, you need to do the same on the other side to keep it level. We'll utilize these principles throughout our solution.

Step-by-Step Solution

Let's tackle the equation 8 + 4 = 2(x - 1) step by step:

Step 1: Simplify Both Sides

Our first step is to simplify both sides of the equation as much as possible. On the left side, we have 8 + 4, which is a simple addition problem. On the right side, we have 2(x - 1), which involves the distributive property. Remember, the distributive property states that a(b + c) = ab + ac. So, we need to multiply the 2 by both 'x' and '-1'.

  • Left side: 8 + 4 = 12
  • Right side: 2(x - 1) = 2 * x + 2 * (-1) = 2x - 2

Now our equation looks like this: 12 = 2x - 2. Much simpler, right?

Step 2: Isolate the Term with 'x'

Next, we want to isolate the term with 'x' (which is 2x) on one side of the equation. To do this, we need to get rid of the '-2' on the right side. We can do this by adding 2 to both sides of the equation. Remember, whatever we do to one side, we must do to the other to keep the equation balanced.

  • 12 + 2 = 2x - 2 + 2
  • 14 = 2x

Now we have 14 = 2x. We're getting closer!

Step 3: Solve for 'x'

Finally, to solve for 'x', we need to get 'x' by itself. It's currently being multiplied by 2. To undo this multiplication, we'll divide both sides of the equation by 2.

  • 14 / 2 = 2x / 2
  • 7 = x

And there you have it! We've solved for x. Our solution is x = 7.

Verifying the Solution

It's always a good idea to verify your solution to make sure it's correct. To do this, we'll substitute x = 7 back into the original equation and see if it holds true.

  • Original equation: 8 + 4 = 2(x - 1)
  • Substitute x = 7: 8 + 4 = 2(7 - 1)
  • Simplify: 12 = 2(6)
  • 12 = 12

The equation holds true! This confirms that our solution, x = 7, is correct.

Common Mistakes to Avoid

When solving algebraic equations, there are a few common mistakes that students often make. Being aware of these pitfalls can help you avoid them.

  • Forgetting the Distributive Property: When dealing with expressions like 2(x - 1), it's crucial to distribute the 2 to both terms inside the parentheses. Forgetting to do so will lead to an incorrect answer.
  • Not Performing Operations on Both Sides: Remember the golden rule of algebra: whatever you do to one side of the equation, you must do to the other. This ensures the equation remains balanced.
  • Incorrectly Combining Like Terms: Be careful when combining like terms. Make sure you're only adding or subtracting terms that have the same variable or are constants.
  • Sign Errors: Pay close attention to signs (positive and negative). A simple sign error can throw off your entire solution.

By being mindful of these common mistakes, you can increase your accuracy and confidence in solving algebraic equations.

Practice Makes Perfect

The best way to master solving for 'x' is through practice. Try working through similar equations to build your skills and understanding. The more you practice, the more comfortable you'll become with the process. You can find plenty of practice problems in textbooks, online resources, or even create your own!

Alternative Methods for Solving

While we've used a step-by-step approach to solve this equation, there are alternative methods you can use. For example, some people prefer to divide both sides of the equation by 2 in the first step. This can simplify the equation earlier on, but it's important to ensure you're comfortable with the division.

Ultimately, the best method is the one that you understand and can apply accurately. Experiment with different approaches to find what works best for you.

Real-World Applications

Algebra isn't just about abstract equations; it has real-world applications in various fields. From engineering and finance to computer science and everyday problem-solving, the ability to solve for unknowns is a valuable skill.

For example, you might use algebraic equations to calculate the cost of materials for a project, determine the speed of a car, or even plan a budget. Understanding the principles of algebra can empower you to tackle a wide range of challenges.

Conclusion

Solving for 'x' in the equation 8 + 4 = 2(x - 1) might have seemed daunting at first, but by breaking it down into manageable steps, we've shown that it's quite achievable. Remember to simplify, isolate the variable, and verify your solution. With practice and a solid understanding of algebraic principles, you'll be solving equations like a pro in no time!

Keep practicing, stay curious, and don't be afraid to ask for help when you need it. Math can be challenging, but it's also incredibly rewarding. You've taken a great step by learning how to solve this equation, and there's a whole world of mathematical knowledge waiting for you to explore!

For further learning and practice, explore resources like Khan Academy's Algebra Section. Happy solving!