Solving For X: 8x - 18 = 2(2x + 3)

by Andrew McMorgan 35 views

Hey math enthusiasts! Today, we're diving deep into solving a classic algebraic equation. We're going to break down the steps to solve for x in the equation 8x - 18 = 2(2x + 3). Whether you're a student tackling homework or just looking to brush up on your algebra skills, this guide will walk you through each step clearly and concisely. Let's get started!

Understanding the Equation

Before we jump into the solution, let's first understand the equation we're dealing with: 8x - 18 = 2(2x + 3). This is a linear equation, which means it involves a variable (x) raised to the power of 1. Our goal is to isolate x on one side of the equation to find its value. To do this, we'll use a series of algebraic manipulations, making sure to maintain the balance of the equation at every step.

Key Concepts in Solving Equations

To effectively solve for x, it's important to understand a few key concepts:

  • Equality: The equals sign (=) signifies that both sides of the equation are equal. Any operation performed on one side must also be performed on the other side to maintain this equality.
  • Inverse Operations: To isolate x, we'll use inverse operations. Addition and subtraction are inverse operations, as are multiplication and division. For example, to undo addition, we subtract; to undo multiplication, we divide.
  • Distributive Property: This property is crucial when dealing with expressions in parentheses. It states that a(b + c) = ab + ac. We'll use this to simplify the equation by removing the parentheses.

With these concepts in mind, we're ready to tackle the equation step-by-step. So, grab your pencils, and let's dive into the solution!

Step-by-Step Solution

Now, let's get to the heart of the matter and solve the equation. We'll break it down into manageable steps, explaining each one along the way. By the end, you'll not only have the answer but also a solid understanding of how we got there.

Step 1: Distribute the 2

Our first step is to simplify the right side of the equation by applying the distributive property. Remember, we need to multiply the 2 outside the parentheses by each term inside the parentheses. Here's how it looks:

2(2x + 3) = (2 * 2x) + (2 * 3) = 4x + 6

So, our equation now becomes:

8x - 18 = 4x + 6

This step is crucial because it removes the parentheses, making the equation easier to work with. By distributing the 2, we've expanded the expression and set the stage for further simplification. This is a common technique in algebra, so mastering it is key to solving various equations.

Step 2: Move the x terms to one side

Next, we want to gather all the terms containing x on one side of the equation. It's generally a good practice to move the smaller x term to the side with the larger x term to avoid dealing with negative coefficients. In this case, we'll subtract 4x from both sides:

8x - 18 - 4x = 4x + 6 - 4x

This simplifies to:

4x - 18 = 6

By subtracting 4x from both sides, we've successfully moved all the x terms to the left side of the equation. This brings us closer to isolating x and finding its value. Remember, maintaining balance is crucial, so we perform the same operation on both sides.

Step 3: Move the constant terms to the other side

Now, let's isolate the x term further by moving the constant terms (the numbers without x) to the right side of the equation. To do this, we'll add 18 to both sides:

4x - 18 + 18 = 6 + 18

This simplifies to:

4x = 24

Adding 18 to both sides effectively cancels out the -18 on the left side, leaving us with just the x term. We're getting closer and closer to our goal! Remember, each step we take is designed to simplify the equation and isolate x.

Step 4: Isolate x by dividing

Finally, we need to isolate x completely. Currently, x is being multiplied by 4. To undo this multiplication, we'll divide both sides of the equation by 4:

(4x) / 4 = 24 / 4

This simplifies to:

x = 6

And there you have it! We've successfully solved for x. By dividing both sides by 4, we've isolated x and found its value. This final step is the culmination of all our previous efforts, and it gives us the solution we were looking for.

The Solution: x = 6

So, after all that algebraic maneuvering, we've arrived at the solution: x = 6. But how do we know if our answer is correct? The best way to check is to substitute the value of x back into the original equation and see if it holds true.

Verification

Let's substitute x = 6 into the original equation: 8x - 18 = 2(2x + 3)

8(6) - 18 = 2(2(6) + 3)

48 - 18 = 2(12 + 3)

30 = 2(15)

30 = 30

The equation holds true! This confirms that our solution, x = 6, is correct. Verification is a crucial step in solving equations, as it ensures that our answer is accurate and that we haven't made any mistakes along the way. It's always a good idea to double-check your work, especially in math!

Common Mistakes to Avoid

Solving equations can be tricky, and it's easy to make mistakes if you're not careful. Here are some common pitfalls to watch out for:

  • Incorrect Distribution: Forgetting to distribute the number outside the parentheses to all terms inside can lead to errors. Always remember to multiply the number by each term.
  • Combining Unlike Terms: You can only combine terms that have the same variable and exponent. For example, you can combine 8x and -4x, but you can't combine 4x and 6.
  • Incorrect Order of Operations: Remember to follow the order of operations (PEMDAS/BODMAS). Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).
  • Not Maintaining Balance: Whatever operation you perform on one side of the equation, you must perform on the other side. Failing to do so will disrupt the balance and lead to an incorrect solution.
  • Sign Errors: Be extra careful when dealing with negative signs. A simple sign error can throw off your entire solution.

By being aware of these common mistakes, you can avoid them and increase your chances of solving equations correctly. Practice makes perfect, so keep working at it!

Practice Problems

Now that we've solved one equation together, it's time for you to put your skills to the test! Here are a few practice problems to try:

  1. 5x + 10 = 3(x - 2)
  2. 2(4x - 1) = 6x + 8
  3. 9x - 15 = 3x + 9

Work through these problems step-by-step, just like we did with the example equation. Remember to distribute, combine like terms, and isolate x. The more you practice, the more confident you'll become in your equation-solving abilities. And don't forget to check your answers by substituting them back into the original equations!

Conclusion

Alright guys, we've reached the end of our equation-solving journey! We've successfully tackled the equation 8x - 18 = 2(2x + 3) and found that x = 6. We walked through each step, from distributing to isolating x, and even verified our solution. Along the way, we discussed key concepts, common mistakes to avoid, and provided some practice problems for you to hone your skills.

Solving equations is a fundamental skill in algebra, and it's essential for tackling more advanced math topics. By mastering the techniques we've covered in this guide, you'll be well-equipped to handle a wide range of equations. So keep practicing, stay curious, and don't be afraid to challenge yourself. You've got this!

Whether you're a student looking to ace your next math test or simply someone who enjoys the thrill of problem-solving, we hope this guide has been helpful. Thanks for joining us, and happy solving!