Solving For *c*: A Step-by-Step Guide

Hey guys! Today, we're diving into a bit of algebra to solve for the variable c in the equation 53 = (3/8)c + 17. Don't worry, it's not as scary as it looks! We'll break it down step by step, making it super easy to understand. So, grab your pencils and let's get started!

Understanding the Equation

Before we jump into solving, let's quickly understand what this equation is telling us. We have a variable, c, which is being multiplied by the fraction 3/8. Then, we're adding 17 to that result, and the final answer is 53. Our goal is to figure out what value of c makes this whole statement true. In algebra, this is a fundamental concept, and mastering it will open doors to solving more complex problems down the road. Think of it like a puzzle – we need to find the missing piece that fits perfectly. This equation represents a linear relationship, and solving for c is like finding a specific point that satisfies this relationship. Equations like these are used everywhere, from calculating distances and speeds to figuring out financial investments. So, understanding how to solve them is a really valuable skill. The beauty of algebra lies in its ability to represent real-world situations with symbols and equations, allowing us to manipulate and solve them. This equation is a perfect example of that, taking a relationship between numbers and a variable and turning it into a solvable problem. We are essentially unwrapping the equation to isolate c, like peeling back the layers of an onion. Each step we take brings us closer to the solution, and the process is actually quite satisfying once you get the hang of it. So, let's dive into the first step of solving for c and see how we can simplify this equation.

Step 1: Isolate the Term with 'c'

The golden rule in solving algebraic equations is to isolate the variable we're interested in. In this case, we want to get the term (3/8)c by itself on one side of the equation. To do this, we need to get rid of the +17 that's hanging out on the right side. The way we do that is by performing the opposite operation. Since we're adding 17, we'll subtract 17 from both sides of the equation. Remember, whatever we do to one side, we must do to the other to keep the equation balanced. This is a crucial concept in algebra – maintaining balance is key to finding the correct solution. Imagine a seesaw; if you add weight to one side, you need to add the same weight to the other side to keep it level. The same principle applies to equations. Subtracting 17 from both sides gives us: 53 - 17 = (3/8)c + 17 - 17. Simplifying this, we get 36 = (3/8)c. Now, we're one step closer! The term with c is more isolated, and we can see the next step more clearly. This step highlights the importance of inverse operations in algebra. Addition and subtraction are inverse operations, and using them strategically allows us to isolate variables and simplify equations. By subtracting 17, we effectively "undid" the addition, bringing us closer to our goal. Isolating the variable is like clearing a path through the jungle – it allows us to see the destination more clearly and move towards it more efficiently. With the term (3/8)c now isolated, we can move on to the next step, which involves dealing with the fraction. Don't worry, it's not as intimidating as it might seem!

Step 2: Get Rid of the Fraction

Now that we have 36 = (3/8)c, we need to deal with that pesky fraction. The c is being multiplied by 3/8, so to get c by itself, we need to do the opposite – we need to divide by 3/8. But here's a handy trick: dividing by a fraction is the same as multiplying by its reciprocal! The reciprocal of 3/8 is 8/3. So, we're going to multiply both sides of the equation by 8/3. Again, remember the balance! What we do to one side, we do to the other. This is a fundamental principle in algebra, ensuring that the equation remains valid throughout the solving process. Think of it as a mathematical dance – every step must be mirrored on both sides to maintain harmony. Multiplying both sides by 8/3 gives us: (8/3) * 36 = (8/3) * (3/8) * c. On the right side, the (8/3) and (3/8) cancel each other out, leaving us with just c. On the left side, we have (8/3) * 36. We can simplify this by multiplying 8 by 36 and then dividing by 3, or we can be clever and notice that 36 is divisible by 3. 36 divided by 3 is 12, so we have 8 * 12. This simplifies the calculation and reduces the risk of errors. Dealing with fractions can sometimes feel tricky, but understanding the concept of reciprocals makes it much easier. Reciprocals are like the inverses of fractions – they undo the effect of the original fraction when multiplied. This technique is a powerful tool in algebra, allowing us to simplify equations and isolate variables more effectively. With the fraction now gone, we're in the home stretch! We've isolated c on one side, and all that's left is to simplify the other side to find its value.

Step 3: Simplify and Find 'c'

Alright, we're in the final stretch! We have (8/3) * 36 = c, which we simplified to 8 * 12 = c. Now, all that's left is to multiply 8 by 12. This is a straightforward multiplication, and you can do it in your head or use a calculator if you prefer. 8 times 12 is 96. So, we have 96 = c. And that's it! We've solved for c. The value of c that makes the original equation true is 96. This final calculation is the culmination of all the steps we've taken, and it's a satisfying moment when we arrive at the solution. It's like reaching the summit of a mountain after a long climb – the view is well worth the effort. Simplifying the expression is a crucial step in algebra, ensuring that we arrive at the most concise and understandable answer. In this case, multiplying 8 by 12 is the final act of simplification, giving us the value of c. We've now successfully navigated the equation, isolated the variable, and found its value. But before we celebrate too much, let's do one more thing – let's check our answer.

Step 4: Check Your Answer

It's always a good idea to check your answer, especially in algebra. This helps ensure that you haven't made any mistakes along the way. To check our answer, we'll substitute c = 96 back into the original equation: 53 = (3/8)c + 17. Replacing c with 96, we get 53 = (3/8) * 96 + 17. Now, we need to simplify the right side of the equation. (3/8) * 96 is the same as 3 * (96/8). 96 divided by 8 is 12, so we have 3 * 12, which is 36. So, the equation becomes 53 = 36 + 17. Adding 36 and 17, we get 53. So, the equation is 53 = 53, which is true! This confirms that our answer, c = 96, is correct. Checking our answer is like proofreading an essay – it's a final step that helps us catch any errors and ensure that our solution is accurate. Substituting the value back into the original equation allows us to verify that it satisfies the equation, giving us confidence in our answer. This step highlights the importance of accuracy in algebra, as even a small mistake can lead to an incorrect solution. By checking our work, we can minimize the risk of errors and ensure that we've arrived at the correct answer. So, we've not only solved for c, but we've also verified our solution, making it a complete and satisfying result.

Conclusion

So, there you have it! We've successfully solved for c in the equation 53 = (3/8)c + 17. We found that c = 96, and we even checked our answer to make sure it's correct. Remember, the key to solving algebraic equations is to isolate the variable you're interested in by performing opposite operations on both sides of the equation. Don't be afraid of fractions – they're just numbers like any other! And always, always check your answer. Algebra might seem tricky at first, but with practice, you'll become a pro in no time. Solving equations is a fundamental skill in mathematics, and it's a skill that will serve you well in many different areas of life. From calculating everyday expenses to understanding complex scientific concepts, algebra provides a powerful framework for problem-solving. So, keep practicing, keep exploring, and keep having fun with math! You've got this! Remember, every equation is just a puzzle waiting to be solved, and with the right tools and techniques, you can crack any code. So, go forth and conquer those equations! You are now equipped to tackle similar problems with confidence and precision. Keep practicing and exploring the world of algebra – it's a fascinating and rewarding journey!

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Solve the equation 53 = (3/8)c + 17 for the variable c.

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Solving for c: A Simple Algebraic Equation Tutorial