Solving The Equation: 4x - 2(x + 1) = 6

by Admin 40 views
Solving the Equation: 4x - 2(x + 1) = 6

Hey guys! Today, we're diving into a classic algebra problem: solving the equation 4x - 2(x + 1) = 6. Don't worry, it's not as scary as it looks! We'll break it down step-by-step so you can master these types of problems. Understanding how to solve linear equations is a fundamental skill in mathematics, and it pops up everywhere from basic algebra to more advanced topics. Plus, it’s super useful in real-life situations where you need to figure out unknown quantities. So, grab your pencils, and let’s get started!

Understanding the Basics of Algebraic Equations

Before we jump into the solution, let's quickly review some key concepts. In an algebraic equation, our main goal is to isolate the variable (in this case, 'x') on one side of the equation. This means we want to get 'x' by itself so we can see its value. To do this, we use a few important principles:

  • The Golden Rule of Algebra: Whatever you do to one side of the equation, you must do to the other side. This keeps the equation balanced.
  • Inverse Operations: To get rid of a term, we use its inverse operation. For example, the inverse of addition is subtraction, and the inverse of multiplication is division.
  • Order of Operations: Remember PEMDAS/BODMAS (Parentheses/Brackets, Exponents/Orders, Multiplication and Division, Addition and Subtraction). This tells us the order in which to perform operations.

With these basics in mind, we're ready to tackle our equation. Keep these principles at the forefront of your mind as we navigate each step. Mastering these basics ensures you have a strong foundation for tackling more complex algebraic problems down the road.

Step-by-Step Solution to 4x - 2(x + 1) = 6

Let's break down the solution into manageable steps.

Step 1: Distribute the -2

The first thing we need to do is get rid of the parentheses. We do this by distributing the -2 across the terms inside the parentheses. Remember, distributing means multiplying the term outside the parentheses by each term inside.

4x - 2(x + 1) = 6

4x - 2 * x - 2 * 1 = 6

This simplifies to:

4x - 2x - 2 = 6

Distributing correctly is crucial because it clears the way for combining like terms, which we’ll do in the next step. Make sure you pay close attention to signs – a negative sign distributed can change the signs of the terms inside the parentheses, which is a common place for errors.

Step 2: Combine Like Terms

Now we have terms with 'x' and constant terms (numbers). Let's combine the like terms on the left side of the equation. We have 4x and -2x, which are like terms because they both contain the variable 'x'.

4x - 2x - 2 = 6

Combining 4x and -2x gives us:

2x - 2 = 6

Combining like terms simplifies the equation, making it easier to work with. This step reduces the number of terms and gets us closer to isolating the variable. Always double-check that you've combined the correct terms and that your signs are accurate.

Step 3: Isolate the Term with 'x'

Our next goal is 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. We use the inverse operation, which is adding 2 to both sides of the equation. Remember the Golden Rule: what we do to one side, we must do to the other.

2x - 2 + 2 = 6 + 2

This simplifies to:

2x = 8

Adding 2 to both sides cancels out the -2 on the left, leaving us with just the term containing 'x'. This step is a key move in isolating our variable and bringing us closer to the final solution. Make sure you perform the same operation on both sides to maintain the equation’s balance.

Step 4: Solve for 'x'

Finally, we need to solve for 'x'. We have 2x = 8. The inverse operation of multiplying by 2 is dividing by 2. So, we divide both sides of the equation by 2.

2x / 2 = 8 / 2

This gives us:

x = 4

And there you have it! We've solved for 'x'. Dividing both sides by the coefficient of 'x' isolates the variable and provides the solution. This final step concludes our journey to solve the equation, showcasing the value of each preceding step.

Checking Our Solution

It's always a good idea to check our solution to make sure we didn't make any mistakes. To do this, we substitute our value for 'x' (which is 4) back into the original equation:

4x - 2(x + 1) = 6

Substitute x = 4:

4(4) - 2(4 + 1) = 6

Now, let's simplify:

16 - 2(5) = 6

16 - 10 = 6

6 = 6

Since both sides of the equation are equal, our solution is correct! Checking your solution is a crucial step in the problem-solving process. It helps you catch any errors and ensures that your answer is accurate. By substituting your solution back into the original equation, you can verify that it satisfies the equation.

Common Mistakes to Avoid

When solving equations like this, there are a few common mistakes you should watch out for:

  • Incorrect Distribution: Make sure you distribute correctly, especially with negative signs. A missed negative sign can throw off the entire solution.
  • Combining Unlike Terms: Only combine like terms. You can't combine terms with 'x' with constant terms.
  • Not Performing Operations on Both Sides: Remember the Golden Rule! Always do the same operation on both sides of the equation.
  • Order of Operations Errors: Stick to PEMDAS/BODMAS to ensure you perform operations in the correct order.

By being aware of these common mistakes, you can avoid them and improve your accuracy when solving equations. Practice and careful attention to detail will help you minimize errors and build confidence in your problem-solving skills.

Practice Problems

Want to test your skills? Try solving these equations:

  1. 3x + 2(x - 1) = 13
  2. 5(x + 2) - x = 14
  3. 2x - 3(x + 4) = -2

Solving these practice problems will reinforce your understanding of the steps involved and help you build confidence in your ability to solve similar equations. Working through a variety of problems will also expose you to different scenarios and challenges, further strengthening your skills.

Conclusion

So, guys, we've successfully solved the equation 4x - 2(x + 1) = 6! We walked through each step, from distributing and combining like terms to isolating the variable and checking our solution. Remember, the key is to take it one step at a time and pay attention to the details.

Solving algebraic equations is a fundamental skill that's essential for success in mathematics and beyond. By mastering these techniques, you're building a strong foundation for more advanced topics and real-world applications. So, keep practicing, stay patient, and you'll become an equation-solving pro in no time! Keep up the great work, and I’ll catch you in the next math adventure!