Solve For R: A Step-by-Step Guide

Hey guys! Let's dive into solving a simple algebraic equation together. We've got this equation: 0.5(r + 2.75) = 3, and our mission, should we choose to accept it, is to find the value of 'r'. Don't worry, it's not as daunting as it looks! We'll break it down step-by-step, making it super easy to follow along. Think of 'r' as a mystery number, and our goal is to unearth its true identity. We'll use some basic algebraic principles to isolate 'r' on one side of the equation, revealing its value. So grab your thinking caps, and let's get started on this mathematical adventure! Remember, the key to success in algebra is to take things slowly and methodically, one step at a time. There's no need to rush, and if you get stuck, just backtrack and review the previous steps. Math is like building a house, each step relies on the ones before it. So let’s lay a solid foundation for understanding this equation. First, we need to understand the order of operations – often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). This will guide us in how to unravel the equation and ultimately solve for our unknown, 'r'. Always remember that whatever operation you perform on one side of the equation, you must also perform on the other side to maintain the balance and integrity of the equation. This is a crucial principle in algebra, ensuring that the equality remains true throughout the solving process.

Step 1: Distribute the 0.5

The first thing we need to tackle is the 0.5 lurking outside the parentheses. This means we're going to distribute, or multiply, the 0.5 by each term inside the parentheses. It's like sharing a pizza – each person inside the parentheses gets a slice! So, we'll multiply 0.5 by 'r' and then 0.5 by 2.75. This gives us: 0.5 * r = 0.5r and 0.5 * 2.75 = 1.375. Now our equation looks like this: 0.5r + 1.375 = 3. See? We've already made progress! Distributing is a fundamental algebraic technique that allows us to simplify expressions and equations. It's like unlocking a door to the next level of solving. By distributing the 0.5, we've essentially removed the parentheses, making the equation easier to manipulate. This step is crucial because it sets the stage for isolating 'r', our ultimate goal. Think of it as preparing the ingredients before you start cooking – you need to have everything ready before you can create the final dish. The same applies to algebra; simplifying the equation is essential before we can solve for the unknown variable. And remember, accuracy is key! Double-check your calculations to ensure that you've multiplied correctly. A small mistake in the distribution step can throw off the entire solution. So, let’s take a moment to appreciate the power of distribution and how it helps us unravel complex equations. Now we’re ready to move on to the next step in our quest to find 'r'.

Step 2: Isolate the Term with 'r'

Our next goal is to get the term with 'r' (0.5r) all by itself on one side of the equation. It's like trying to separate a single sock from a pile of laundry. To do this, we need to get rid of the 1.375 that's hanging out with our 0.5r. Since 1.375 is being added, we'll do the opposite – we'll subtract 1.375 from both sides of the equation. Remember, whatever we do to one side, we have to do to the other to keep things balanced! So, we subtract 1.375 from both sides: 0. 5r + 1.375 - 1.375 = 3 - 1.375. This simplifies to 0.5r = 1.625. Awesome! We're one step closer to finding 'r'. Isolating the variable term is a core strategy in solving algebraic equations. It's like building a fortress around the variable, protecting it from external influences. By subtracting 1.375 from both sides, we've effectively neutralized its presence on the left side of the equation, leaving 0.5r exposed and ready for the final step. This step highlights the importance of inverse operations – using the opposite operation to undo an operation. Since addition and subtraction are inverse operations, we used subtraction to undo the addition of 1.375. This principle applies to other operations as well, such as multiplication and division, which we'll encounter in the next step. And again, precision is paramount. Make sure you subtract the same value from both sides of the equation and that your calculations are accurate. A small error here can lead to an incorrect solution. So, let’s take a moment to recognize the significance of isolating the variable term and how it paves the way for solving for the unknown. We're on the home stretch now!

Step 3: Solve for 'r'

Now for the grand finale! We have 0.5r = 1.625. This means 0.5 times 'r' equals 1.625. To find 'r', we need to undo the multiplication. And how do we do that? By dividing! We'll divide both sides of the equation by 0.5. This is like splitting a bill – we're dividing the total amount by the number of people. So, we divide both sides by 0.5: 0.5r / 0.5 = 1.625 / 0.5. This simplifies to r = 3.25. Huzzah! We've found 'r'! It's 3.25. Solving for the variable is the culmination of all our efforts. It's the moment when the mystery number is revealed, and the puzzle is complete. By dividing both sides of the equation by 0.5, we've effectively isolated 'r' and determined its value. This step reinforces the importance of inverse operations and how they help us unravel equations. Just as we used subtraction to undo addition, we used division to undo multiplication. This consistent application of inverse operations is a hallmark of algebraic problem-solving. And, of course, accuracy is crucial in this final step. Make sure you divide both sides of the equation by the correct value and that your calculation is precise. A small error here can negate all the previous work. So, let’s take a moment to celebrate our success in solving for 'r'. We've conquered the equation and discovered that 'r' is indeed 3.25. Now we know the answer, but let's not stop there. Let's take one final step to ensure our solution is correct.

Step 4: Verify the Solution

To be absolutely sure we've nailed it, let's check our answer. We'll plug our value for 'r' (3.25) back into the original equation and see if it holds true. It's like double-checking your work before submitting a test – you want to make sure everything adds up! So, let's substitute 3.25 for 'r' in the original equation: 0.5(3.25 + 2.75) = 3. First, we'll simplify inside the parentheses: 3.25 + 2.75 = 6. Now we have: 0.5(6) = 3. And finally, 0.5 * 6 = 3. So, 3 = 3. It checks out! Our solution is correct. Verifying the solution is a vital step in the problem-solving process. It's like the final seal of approval, confirming that our answer is accurate and that we've successfully solved the equation. By substituting our calculated value of 'r' back into the original equation, we can ensure that the equation remains balanced and that both sides are equal. This process not only validates our solution but also helps us identify any potential errors in our calculations. If the equation doesn't hold true after substitution, it indicates that there might be a mistake in our steps, and we need to revisit our work to pinpoint the error. This self-checking mechanism is invaluable in algebra and beyond. It fosters a habit of carefulness and attention to detail, which are essential skills in any field. So, let’s take a moment to appreciate the importance of verifying the solution and how it strengthens our confidence in our answer. We've not only solved the equation but also confirmed that our solution is correct. That’s a double win!

Conclusion

And there you have it! We've successfully solved for 'r' in the equation 0.5(r + 2.75) = 3, and we found that r = 3.25. We even double-checked our work to make sure we were right on the money. Remember, solving algebraic equations is all about breaking them down into manageable steps, using inverse operations, and keeping the equation balanced. With practice, you'll become a pro at solving for any variable! So, the next time you encounter an equation like this, don't be intimidated. Remember the steps we've covered, and you'll be well on your way to finding the solution. Keep practicing, keep exploring, and keep those math muscles strong! Solving equations is a fundamental skill in mathematics, and it opens doors to more advanced concepts and applications. From physics to engineering to economics, equations are used to model and solve real-world problems. So, the ability to confidently solve equations is a valuable asset in any field. And remember, math is not just about numbers and symbols; it's about logical thinking, problem-solving, and critical analysis. These are skills that will serve you well in all aspects of life. So, embrace the challenge, enjoy the journey, and never stop learning. The world of mathematics is vast and fascinating, and there's always something new to discover. So, let’s celebrate our success in solving this equation and continue our exploration of the wonderful world of math! You've got this!