Solving For 'a': Unveiling The Value In √ (6a+1) = 5

Hey guys! Let's dive into a fun little math problem. We're going to solve for 'a' in the equation √ (6a+1) = 5. 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. This is a classic example of how to isolate a variable in an equation involving a square root. Understanding this process is crucial for various mathematical concepts, and it's a great way to brush up on your algebra skills. The equation we're tackling, √ (6a+1) = 5, might seem a bit intimidating at first glance, but with the right approach, it's totally manageable. Our goal is to find the value of 'a' that makes this equation true. Think of it like a puzzle where we have to uncover the missing piece. We'll utilize the properties of square roots and algebraic manipulation to get to the answer. Remember, the key to solving any equation is to isolate the variable, and that's precisely what we're going to do here. We'll start by eliminating the square root, and then we'll work our way through the rest of the equation until we have the value of 'a'. By the time we're done, you'll be able to solve similar equations with confidence! So, let's roll up our sleeves and get started. This will not only show you how to solve this equation, but it will also give you a better understanding of how equations work and how to manipulate them to find the unknown values. The first step involves eliminating the square root, which will open the door to solving for 'a' directly. The following steps should be followed, so be patient, and we will solve it together.

Step 1: Squaring Both Sides to Eliminate the Square Root

Alright, first things first, we need to get rid of that pesky square root. And how do we do that? By squaring both sides of the equation! Squaring both sides is like performing a perfectly legal mathematical magic trick. It's a fundamental principle: whatever you do to one side of the equation, you must do to the other to keep things balanced. So, we'll square both sides of our equation: (√ (6a+1))² = 5². This might look a bit intimidating, but trust me, it's simpler than it seems. The act of squaring the square root does exactly what we want: it removes the square root sign, leaving us with 6a+1 on the left side. On the right side, we have 5² which is simply 25. So, our equation transforms into 6a + 1 = 25. See? We've already simplified things quite a bit. This step is about getting rid of the square root, which is the main roadblock in solving for 'a'. This allows us to work with a much simpler equation. Remember, squaring both sides doesn't change the equation's core meaning; it just makes it easier to handle. This process is crucial because it allows us to work with a linear equation, which is much easier to solve than an equation involving a square root. This step is the key to unlocking the solution. The core rule here is that when you square a square root, they cancel each other out, leaving only the original value under the radical. It's a fundamental principle of algebra, and understanding it is key to mastering this type of equation. It sets the stage for the next steps, where we'll isolate 'a'.

Step 2: Isolating the Term with 'a'

Now that we've banished the square root, it's time to focus on isolating the term containing 'a'. We want to get '6a' by itself on one side of the equation. To do this, we need to get rid of that '+1' that's hanging around. How do we do that? By subtracting 1 from both sides of the equation. Remember, we need to keep things balanced. So, our equation 6a + 1 = 25 becomes 6a + 1 - 1 = 25 - 1. This leaves us with a simplified equation: 6a = 24. This step is all about getting closer to the solution. By removing the constant term, we've set the stage to isolate the variable 'a'. Think of it like peeling away layers of the equation to get to the core: the value of 'a'. This process adheres to the fundamental principle of equation solving: performing the same operation on both sides of the equation maintains the equality. Therefore, by subtracting 1 from both sides, we ensure that the equation remains valid while moving us closer to our goal: the solution for 'a'. In this step, we've reduced the equation to a form where 'a' is only multiplied by a number (6). This is the key step in isolating 'a'. This step is a direct application of algebraic principles, demonstrating how to manipulate equations to isolate a variable. Understanding this step will enable you to solve various algebraic problems. This step represents a crucial shift in the equation, bringing us closer to the final solution. The goal is to isolate the variable 'a' and determine its value.

Step 3: Solving for 'a'

Almost there, guys! We now have the equation 6a = 24. Our final step is to solve for 'a'. Right now, 'a' is being multiplied by 6. To isolate 'a', we need to do the opposite of multiplication, which is division. So, we'll divide both sides of the equation by 6. This gives us (6a) / 6 = 24 / 6. The 6's on the left side cancel each other out, leaving us with 'a' by itself. On the right side, 24 divided by 6 is 4. Therefore, our solution is a = 4. Congratulations! You've successfully solved for 'a'. This is the final step, and we've successfully isolated the variable. By dividing both sides of the equation by the coefficient of 'a', we've found its value. This step showcases the power of inverse operations in solving equations. Dividing both sides by 6, we've effectively undone the multiplication and revealed the value of 'a'. This process underscores the fundamental principles of algebra. This step is where we find the final answer. It is a simple division that leads us to the value of 'a'. The process, in this step, demonstrates how to solve for an unknown variable in an equation. This step is a critical component of the whole process. By dividing both sides by the coefficient of 'a', we isolate the variable and reveal its numerical value. This step is a straightforward application of mathematical principles. This is where we celebrate the solution.

Step 4: Verification of the Solution

Always double-check your work, people! It's super important to verify your solution. Let's substitute a = 4 back into the original equation: √ (6a+1) = 5. Replacing 'a' with 4, we get √ (6*4 + 1) = 5. Simplifying this, we have √ (24 + 1) = 5, which gives us √25 = 5. And hey, the square root of 25 is indeed 5! So, 5 = 5. This confirms that our solution, a = 4, is correct. Verifying the solution is an essential part of the problem-solving process. This confirms that you've arrived at the correct answer. This step is a chance to ensure there were no errors during the problem-solving process. Verification gives us confidence in the solution's accuracy. This step highlights the importance of accuracy in mathematical problem-solving. This verification ensures the solution matches the original equation. This is not only a good practice but also a valuable tool in ensuring the solution's validity. This verification guarantees the accuracy of the solution, which reinforces the understanding of the equation-solving process.

Conclusion: Mastering the Equation

There you have it, folks! We've successfully solved for 'a' in the equation √ (6a+1) = 5. We started with a seemingly complex equation and broke it down into manageable steps: squaring both sides to eliminate the square root, isolating the 'a' term, solving for 'a', and verifying the solution. Each step was designed to simplify the equation and get us closer to our final answer. The key takeaways from this exercise are understanding how to manipulate equations, how to work with square roots, and the importance of verifying your solutions. Now you can use these skills to solve other similar equations with confidence. Remember, practice makes perfect. Keep practicing, and you'll be solving equations like a pro in no time! Keep in mind that understanding the properties of exponents and radicals is crucial. This will help you to solve more complex equations. Understanding each step, from eliminating the square root to verifying the solution, provides a comprehensive grasp of the equation-solving process. Each step played a crucial role in reaching the final result, highlighting the importance of a systematic approach in mathematics. The principles used here can be applied to many other equations and scenarios. Keep practicing, and you'll become more and more proficient with these types of problems. Remember, the journey is just as important as the destination. Embrace the process, and enjoy the satisfaction of solving the equation! This exercise gives us a foundation for solving more complex equations, making us more confident and skilled in mathematics. Keep these steps in mind, and you will be able to solve many other equations in the future!