Solving 12 = (5/6)c - 8 A Step-by-Step Guide

Hey guys! Ever feel like you're staring at an equation and it's staring right back, daring you to solve it? Don't worry, we've all been there. Today, we're going to break down a common type of equation step-by-step, making it super easy to understand. We'll be focusing on the equation 12 = (5/6)c - 8, and by the end of this article, you'll be solving similar problems like a pro. We're not just going to give you the answer; we're going to show you why each step works, so you can tackle any equation that comes your way. This isn't just about memorizing steps; it's about understanding the underlying principles of algebra. So, grab a pen and paper, and let's dive into the world of equation solving! We'll cover everything from isolating the variable to simplifying fractions, ensuring you have a solid grasp of the fundamentals. Whether you're a student tackling homework or just someone looking to brush up on their math skills, this guide is for you. Let’s make math less intimidating and more, dare I say, fun!

Understanding the Basics of Algebraic Equations

Before we jump into solving our specific equation, let's quickly review some key concepts in algebra. Think of an equation like a balanced scale. The equals sign (=) is the fulcrum, the center point. Whatever you do to one side of the equation, you must do to the other side to keep the scale balanced. This principle is the golden rule of equation solving! The variable, in our case 'c', is the unknown value we're trying to find. Our goal is to isolate this variable, meaning we want to get it all by itself on one side of the equation. To do this, we use inverse operations. Addition and subtraction are inverse operations; multiplication and division are inverse operations. This means that if an equation has a term being subtracted, we can add to both sides to “undo” the subtraction. Similarly, if a term is being multiplied, we can divide both sides to undo the multiplication. These are the fundamental tools in our algebraic toolbox, and they're crucial for solving any equation. Remember, the key is to maintain balance and systematically work towards isolating the variable. It’s like peeling an onion, you remove layer by layer until you get to the core. Each step we take is designed to simplify the equation and bring us closer to the solution. And trust me, the feeling of solving a complex equation is incredibly rewarding!

Step 1 Adding 8 to Each Side of the Equation

Okay, let's get started with our equation: 12 = (5/6)c - 8. The first thing we want to do is isolate the term with the variable, which is (5/6)c. Notice that there's a "- 8" on the same side as this term. To get rid of it, we need to do the inverse operation, which is adding 8. But remember the golden rule: what we do to one side, we must do to the other! So, we add 8 to both sides of the equation. This gives us: 12 + 8 = (5/6)c - 8 + 8. Now, let's simplify. On the left side, 12 + 8 equals 20. On the right side, -8 + 8 cancels out, leaving us with just (5/6)c. So, our equation now looks like this: 20 = (5/6)c. See how much simpler it is already? We've eliminated the constant term on the right side, bringing us one step closer to isolating 'c'. Adding the same value to both sides is a fundamental algebraic technique, and it's essential for maintaining the equation's balance. Think of it as shifting weight on our balanced scale – we're adding the same weight to each side, so the scale remains level. This step might seem simple, but it's a crucial building block for solving more complex equations.

Step 2 Simplifying the Equation

After adding 8 to both sides, our equation is now 20 = (5/6)c. This step is about making sure everything is as clean and clear as possible before we move on. In this case, the left side is already simplified (20 is a simple number), and the right side has a fraction multiplying our variable. We've successfully eliminated the constant term on the right side, making it easier to isolate 'c'. Simplification is a continuous process in equation solving. It's about making the equation more manageable and easier to work with. Each simplification step brings us closer to the solution. Now, with our simplified equation, we can clearly see that 'c' is being multiplied by the fraction 5/6. Our next step will involve dealing with this fraction, and we'll do that by using the inverse operation of multiplication, which is division. But instead of dividing by a fraction, we'll use a clever trick that makes things even easier: multiplying by the reciprocal. Keep following along, and you'll see how it works!

Step 3 Multiplying by the Reciprocal to Isolate 'c'

We've reached the final step in isolating 'c'! Our equation is 20 = (5/6)c. To get 'c' by itself, we need to undo the multiplication by the fraction 5/6. The best way to do this is by multiplying both sides of the equation by the reciprocal of 5/6. The reciprocal of a fraction is simply the fraction flipped upside down. So, the reciprocal of 5/6 is 6/5. Now, we multiply both sides of the equation by 6/5: 20 * (6/5) = (5/6)c * (6/5). Let's look at what happens on each side. On the left side, we have 20 multiplied by 6/5. We can think of 20 as 20/1, so we're multiplying (20/1) * (6/5). Before we multiply, we can simplify by dividing 20 and 5 by their greatest common factor, which is 5. This gives us (4/1) * (6/1), which equals 24. On the right side, we have (5/6)c multiplied by (6/5). When you multiply a fraction by its reciprocal, the result is always 1. So, (5/6) * (6/5) equals 1, and we're left with just 1 * c, which is simply c. Therefore, our equation now looks like this: 24 = c. We've done it! We've successfully isolated 'c' and found its value. This step highlights the power of using reciprocals to eliminate fractions in equations. It's a technique that can save you a lot of time and effort compared to dividing by a fraction. Multiplying by the reciprocal is like using a mathematical key to unlock the variable from the fractional coefficient.

Solution c = 24

So, after all the steps, we've arrived at our solution: c = 24. This means that the value of 'c' that makes the original equation true is 24. But we're not done yet! It's always a good idea to check our answer to make sure we haven't made any mistakes along the way. To check our solution, we'll substitute 24 back into the original equation: 12 = (5/6)c - 8. Replacing 'c' with 24, we get: 12 = (5/6) * 24 - 8. Now, let's simplify. (5/6) * 24 is the same as (5 * 24) / 6, which equals 120 / 6, which equals 20. So, our equation becomes: 12 = 20 - 8. Finally, 20 - 8 equals 12, so we have: 12 = 12. This is a true statement! This confirms that our solution, c = 24, is correct. Checking your work is a crucial part of problem-solving in mathematics. It gives you confidence in your answer and helps you catch any errors you might have made. It’s like proofreading a paper before you submit it – you want to make sure everything is accurate and makes sense. Plus, the satisfaction of knowing you've solved a problem correctly is a great feeling!

Key Takeaways and Tips for Solving Equations

Alright guys, we've successfully solved the equation 12 = (5/6)c - 8 and found that c = 24. But more importantly, we've learned a powerful process for solving equations in general. Let's recap some key takeaways and tips that will help you tackle any equation that comes your way:

• The Golden Rule: Always remember to do the same thing to both sides of the equation to maintain balance.
• Inverse Operations: Use inverse operations to isolate the variable. Addition and subtraction are inverses, as are multiplication and division.
• Simplify: Simplify each side of the equation as much as possible before moving on to the next step.
• Reciprocals: Multiplying by the reciprocal is a great way to eliminate fractions.
• Check Your Work: Always substitute your solution back into the original equation to make sure it's correct.

Solving equations is like building a house – you start with the foundation (understanding the basics) and then add the layers (steps) until you reach the finished product (the solution). Each step is important, and by following these tips, you'll be well-equipped to solve a wide range of equations. Remember, practice makes perfect! The more equations you solve, the more comfortable and confident you'll become. So, don't be afraid to tackle challenging problems. With a little effort and these strategies, you'll be solving equations like a math whiz in no time!

Practice Problems to Boost Your Skills

Now that we've walked through the solution and discussed some key strategies, it's time to put your skills to the test! Solving equations is a skill that gets better with practice, so let's try a few more problems. I encourage you to work through these on your own, using the steps and tips we've discussed. Remember, the goal isn't just to get the right answer, but to understand the process. So, show your work and think carefully about each step. Here are a couple of practice problems for you to try:

1. 3x + 5 = 14
2. (2/3)y - 1 = 7

For each problem, try to identify the variable, isolate it using inverse operations, and then check your answer. Don't worry if you make mistakes – that's part of the learning process! If you get stuck, revisit the steps we discussed earlier or ask for help. The key is to keep practicing and building your confidence. Think of these practice problems as workouts for your math brain. The more you exercise it, the stronger it will become. And just like any skill, the more you practice solving equations, the more natural and intuitive it will feel. So, grab a pen and paper, and let's get those brain muscles working!

By mastering the art of solving equations, you're not just learning a math skill; you're developing valuable problem-solving abilities that will benefit you in all areas of life. So, keep practicing, keep learning, and keep challenging yourself. You've got this!