Solving Equations: A Step-by-Step Guide

Hey guys! Ever find yourself staring blankly at an equation, wondering where to even begin? Don't worry, we've all been there. Today, we're going to break down the process of solving equations, using the example 4(-2k - 8) = 6(k + 18). This guide will provide you with a step-by-step approach, so you can tackle any equation with confidence. Let's dive in and make math a little less scary, shall we?

Understanding the Basics of Equations

Before we jump into the specific equation, let's quickly recap some fundamental concepts about equations. In essence, an equation is a mathematical statement that asserts the equality of two expressions. Think of it like a balanced scale; both sides must weigh the same. The goal of solving an equation is to isolate the variable – in our case, 'k' – on one side of the equation. This means we want to get 'k' all by itself, so we know its value.

To achieve this, we use inverse operations. Inverse operations are operations that "undo" each other. For example, addition and subtraction are inverse operations, and so are multiplication and division. We apply these operations to both sides of the equation to maintain the balance. Remember, whatever you do to one side, you must do to the other! This ensures that the equation remains equal. This fundamental principle is the key to solving any equation, no matter how complex it may seem at first glance. By understanding inverse operations and maintaining balance, you'll be well-equipped to tackle a wide range of mathematical problems. Keep this in mind as we move forward and break down the equation step-by-step. Mastering these basics is crucial for success in algebra and beyond, so let's get started!

Step 1: Distribute the Numbers

The first step in solving our equation, 4(-2k - 8) = 6(k + 18), is to simplify both sides by distributing the numbers outside the parentheses. This means we'll multiply the number outside each set of parentheses by each term inside. On the left side, we have 4 multiplied by (-2k - 8). So, we'll multiply 4 by -2k, which gives us -8k, and then multiply 4 by -8, which gives us -32. This transforms the left side of the equation into -8k - 32.

On the right side, we have 6 multiplied by (k + 18). Similarly, we'll multiply 6 by k, resulting in 6k, and then multiply 6 by 18, which gives us 108. So, the right side of the equation becomes 6k + 108. Now, our equation looks much simpler: -8k - 32 = 6k + 108. Distributing the numbers is a crucial step because it eliminates the parentheses, making it easier to combine like terms and isolate the variable. By performing this step carefully, we've cleared the path for the next steps in solving the equation. Remember, the key is to multiply correctly and pay attention to the signs. This careful distribution sets the foundation for solving the equation effectively. Keep in mind, distribution is a common technique used in various algebraic problems, so mastering this skill is essential for your mathematical journey. Let’s move on to the next step and see how we can further simplify the equation!

Step 2: Combine Like Terms (Moving Variables)

Now that we've distributed the numbers, our equation looks like this: -8k - 32 = 6k + 108. The next step is to combine like terms. In this case, we want to get all the 'k' terms on one side of the equation and all the constant terms on the other side. It doesn't matter which side we choose for the 'k' terms, but let's aim for the side that will result in a positive coefficient for 'k' to avoid dealing with negative numbers unnecessarily.

To move the -8k term from the left side to the right side, we'll add 8k to both sides of the equation. Remember, whatever we do to one side, we must do to the other to maintain balance. Adding 8k to both sides gives us: -8k - 32 + 8k = 6k + 108 + 8k. Simplifying this, the -8k and +8k on the left side cancel out, leaving us with -32. On the right side, 6k + 8k combines to 14k, so we have 14k + 108. Our equation now looks like this: -32 = 14k + 108. We've successfully moved the variable terms to one side. This step is essential because it brings us closer to isolating 'k' and finding its value. By carefully adding the same term to both sides, we've maintained the equation's balance while simplifying it. This process of combining like terms is a fundamental technique in algebra and is used extensively in solving various types of equations. With the variable terms on one side, we can now focus on moving the constant terms to the other side. Let's proceed to the next step and continue our journey towards finding the solution!

Step 3: Combine Like Terms (Moving Constants)

We've reached the point where our equation is -32 = 14k + 108. Now, we need to isolate the 'k' term further by moving the constant term, 108, to the left side of the equation. To do this, we'll subtract 108 from both sides. Remember, maintaining balance is crucial, so whatever we do to one side, we must do to the other.

Subtracting 108 from both sides gives us: -32 - 108 = 14k + 108 - 108. On the left side, -32 - 108 equals -140. On the right side, the +108 and -108 cancel each other out, leaving us with just 14k. Our equation now looks like this: -140 = 14k. We're getting closer to isolating 'k'! By subtracting the constant from both sides, we've successfully separated the variable term from the constant term. This step is a critical part of the process, as it sets us up for the final step of solving for 'k'. The key here was to identify the constant term and perform the inverse operation (subtraction) on both sides. This ensures that the equation remains balanced while simplifying it. With the variable term now isolated on one side, we are just one step away from finding the value of 'k'. Let's move on to the final step and bring this equation to a satisfying conclusion!

Step 4: Isolate the Variable

Our equation has now been simplified to -140 = 14k. To finally solve for 'k', we need to isolate it completely. This means getting 'k' by itself on one side of the equation. Currently, 'k' is being multiplied by 14. To undo this multiplication, we'll perform the inverse operation: division. We'll divide both sides of the equation by 14.

Dividing both sides by 14 gives us: -140 / 14 = 14k / 14. On the left side, -140 divided by 14 is -10. On the right side, 14k divided by 14 simplifies to k. So, we have -10 = k. Therefore, the solution to the equation is k = -10. We've done it! By dividing both sides by the coefficient of 'k', we've successfully isolated the variable and found its value. This final step is the culmination of all the previous steps, where we carefully simplified the equation and maintained balance. The key here was to identify the operation being performed on the variable (multiplication) and then perform the inverse operation (division) on both sides. Congratulations! You've mastered the process of solving this equation. Let's move on to the next crucial step: verifying our solution.

Step 5: Verify the Solution

We've found that k = -10 is the solution to our equation, but it's always a good idea to double-check our work to make sure we haven't made any mistakes. To verify our solution, we'll substitute -10 for 'k' in the original equation: 4(-2k - 8) = 6(k + 18). If both sides of the equation are equal after the substitution, then our solution is correct.

Let's substitute -10 for 'k': 4(-2(-10) - 8) = 6(-10 + 18). Now, we'll simplify each side. On the left side, -2(-10) is 20, so we have 4(20 - 8). Simplifying further, 20 - 8 is 12, so we have 4(12), which equals 48. So, the left side of the equation is 48. On the right side, -10 + 18 is 8, so we have 6(8), which equals 48. Therefore, the right side of the equation is also 48. Since both sides of the equation are equal (48 = 48), our solution k = -10 is correct! Verifying the solution is a crucial step in problem-solving, as it gives us confidence in our answer and helps us catch any errors we might have made along the way. By substituting the value back into the original equation, we've ensured that our solution satisfies the equation. This step reinforces the accuracy of our work and solidifies our understanding of the problem-solving process. Great job! You've successfully solved and verified the equation. Now, let's wrap things up with a summary of the key steps we've learned.

Conclusion

Alright guys, we've successfully tackled the equation 4(-2k - 8) = 6(k + 18) and found that k = -10. Remember, solving equations is all about following a systematic approach. We started by distributing the numbers to eliminate parentheses, then combined like terms to simplify the equation. Next, we isolated the variable by performing inverse operations on both sides. Finally, we verified our solution to ensure accuracy.

The key takeaways from this exercise are:

1. Distribution: Multiply the number outside the parentheses by each term inside.
2. Combining Like Terms: Group variable terms on one side and constant terms on the other.
3. Inverse Operations: Use opposite operations (addition/subtraction, multiplication/division) to isolate the variable.
4. Verification: Substitute the solution back into the original equation to check your work.

By mastering these steps, you'll be well-equipped to solve a wide range of algebraic equations. Keep practicing, and you'll become a pro in no time! Remember, math can be challenging, but with a clear strategy and a bit of perseverance, you can conquer any equation that comes your way. Keep up the great work, and happy solving! This is just the beginning of your mathematical journey, and I'm confident that you'll continue to grow and excel. Remember to break down complex problems into smaller, manageable steps, and always verify your solutions. With practice and dedication, you'll be able to approach any mathematical challenge with confidence and skill.