Solve 5/6a + 2 = 1/3a - 4: A Step-by-Step Guide

Hey there, math enthusiasts! Ever stumbled upon an equation that looks like it’s written in a foreign language? Equations can seem intimidating, but trust me, they're just puzzles waiting to be solved. Today, we’re going to dissect a particular equation: 5/6a + 2 = 1/3a - 4. This might look complex at first glance, but with a step-by-step approach, we'll break it down and conquer it together. So, grab your pencils, notebooks, and let's dive into the world of algebraic equations!

Understanding the Equation

Before we jump into solving, let's make sure we understand what the equation is telling us. The equation 5/6a + 2 = 1/3a - 4 is an algebraic equation, which means it involves variables (in this case, 'a') and constants connected by mathematical operations. Our goal is to find the value of 'a' that makes the equation true. Think of it like a balancing scale: the left side must equal the right side. To achieve this balance, we'll use various algebraic techniques to isolate 'a' on one side of the equation.

The key to solving any equation is to perform the same operations on both sides. This ensures the equation remains balanced. Whether we add, subtract, multiply, or divide, we must do it consistently on both the left-hand side (LHS) and the right-hand side (RHS). This principle is fundamental to maintaining equality and arriving at the correct solution.

To get started, it's often helpful to identify the different components of the equation. We have fractions (5/6 and 1/3), the variable 'a', and constants (+2 and -4). Our strategy will involve manipulating these components to group like terms together. This typically means getting all the terms with 'a' on one side of the equation and all the constant terms on the other side. Once we achieve this separation, we can simplify each side and ultimately solve for 'a'. Remember, the process might seem a bit like a maze, but with each step, we're getting closer to the solution. So, let’s roll up our sleeves and start maneuvering those terms!

Step-by-Step Solution

Step 1: Eliminate Fractions

Fractions can sometimes make equations look messier than they are. The first step in solving 5/6a + 2 = 1/3a - 4 is to eliminate these fractions. How do we do that? We find the least common multiple (LCM) of the denominators. In this case, the denominators are 6 and 3. The LCM of 6 and 3 is 6. So, we'll multiply both sides of the equation by 6. This might sound like a big step, but it's a clever trick to clear out those fractions and make the equation much easier to handle.

Multiplying both sides by 6, we get:

6 * (5/6a + 2) = 6 * (1/3a - 4)

Now, we distribute the 6 on both sides:

(6 * 5/6a) + (6 * 2) = (6 * 1/3a) - (6 * 4)

Simplifying each term, we have:

5a + 12 = 2a - 24

See how much cleaner the equation looks now? By eliminating the fractions, we've transformed a potentially daunting equation into a more manageable form. This is a common technique in algebra, and mastering it can make solving equations significantly smoother. We've essentially cleared the path for the next steps in our solution. So, with the fractions out of the way, we can now focus on isolating the variable 'a'.

Step 2: Group Like Terms

Now that we've eliminated the fractions, the equation 5a + 12 = 2a - 24 is much easier to handle. The next step is to group like terms. This means we want to get all the terms with 'a' on one side of the equation and all the constant terms on the other side. This is a crucial step in isolating 'a' and ultimately finding its value. Think of it as sorting your laundry – we're grouping similar items together to make the process more organized.

To move the '2a' term from the right side to the left side, we subtract '2a' from both sides of the equation:

5a + 12 - 2a = 2a - 24 - 2a

This simplifies to:

3a + 12 = -24

Next, we want to move the constant term '+12' from the left side to the right side. To do this, we subtract 12 from both sides:

3a + 12 - 12 = -24 - 12

This simplifies to:

3a = -36

We've successfully grouped the like terms! All the 'a' terms are now on the left side, and all the constant terms are on the right side. This brings us one step closer to solving for 'a'. By carefully moving terms across the equals sign while maintaining balance, we've created a simplified equation that's much easier to solve. The next step is the final push to isolate 'a' completely.

Step 3: Isolate the Variable

We're almost there! We've simplified the equation to 3a = -36. Now, the final step is to isolate the variable 'a'. This means we need to get 'a' by itself on one side of the equation. Currently, 'a' is being multiplied by 3. To undo this multiplication, we perform the inverse operation: division. We'll divide both sides of the equation by 3. This will effectively separate 'a' from its coefficient and reveal its value.

Dividing both sides by 3, we get:

(3a) / 3 = (-36) / 3

Simplifying, we find:

a = -12

And there we have it! We've successfully solved for 'a'. The value of 'a' that makes the original equation true is -12. This final step of isolating the variable is the culmination of all our previous efforts. By understanding inverse operations and applying them correctly, we can unravel even the most complex-looking equations. So, with 'a' standing alone, we've conquered the puzzle and found our solution.

Checking the Solution

Congratulations! We've found a solution, but before we declare victory, it's always a good idea to check our work. This step ensures that our answer is correct and that we haven't made any mistakes along the way. Checking our solution involves substituting the value we found for 'a' back into the original equation. If both sides of the equation are equal after the substitution, then our solution is correct.

Our original equation was:

5/6a + 2 = 1/3a - 4

We found that a = -12. Let's substitute this value into the equation:

5/6 * (-12) + 2 = 1/3 * (-12) - 4

Now, we simplify each side. On the left side:

(5/6 * -12) + 2 = -10 + 2 = -8

On the right side:

(1/3 * -12) - 4 = -4 - 4 = -8

Both sides of the equation equal -8! This confirms that our solution, a = -12, is indeed correct. Checking our solution is like the final piece of a jigsaw puzzle – it completes the picture and gives us confidence in our answer. By verifying our work, we not only ensure accuracy but also reinforce our understanding of the equation-solving process. So, with a confirmed solution, we can confidently move on to the next mathematical challenge.

Common Mistakes to Avoid

When solving equations, it's easy to make small errors that can lead to incorrect answers. Let's go over some common mistakes to watch out for. One frequent mistake is not performing the same operation on both sides of the equation. Remember, the equation is like a balance scale, and any operation must be applied equally to both sides to maintain that balance. For example, if you subtract a number from one side, you must subtract the same number from the other side.

Another common error is with the signs. Be careful when dealing with negative numbers! A misplaced negative sign can completely change the outcome. When moving terms across the equals sign, remember to change their signs. For instance, if you move a positive term from one side to the other, it becomes negative, and vice versa.

Fractions can also be a source of errors. When eliminating fractions, make sure you multiply every term on both sides of the equation by the LCM. It's easy to forget a term, especially if there are many terms in the equation. Double-check your work to ensure you've multiplied every term correctly.

Finally, always simplify your equation before moving on to the next step. Simplifying makes the equation easier to work with and reduces the chances of making errors. Combine like terms and reduce fractions whenever possible. By being mindful of these common mistakes and taking the time to check your work, you can improve your accuracy and become a more confident equation solver. Remember, practice makes perfect, so keep solving and learning!

Conclusion

Well, guys, we've reached the end of our journey through the equation 5/6a + 2 = 1/3a - 4. We started with what might have seemed like a daunting problem, but we broke it down into manageable steps and conquered it together! We learned how to eliminate fractions, group like terms, isolate the variable, and even check our solution. Each step is a valuable tool in your algebraic arsenal, and mastering these techniques will empower you to tackle a wide range of equations. Remember, solving equations is like learning a new language – it takes practice, patience, and a willingness to make mistakes and learn from them.

The key takeaways from this exercise are the importance of maintaining balance in an equation, the power of inverse operations, and the necessity of checking your work. By performing the same operations on both sides, we ensure that the equation remains true. Inverse operations allow us to undo mathematical operations and isolate the variable. And checking our solution gives us confidence in our answer and helps us catch any errors.

So, the next time you encounter an equation that looks challenging, remember the steps we've covered. Break it down, stay organized, and don't be afraid to make mistakes. Every mistake is a learning opportunity. Keep practicing, keep exploring, and most importantly, keep having fun with math! You've got this!