Solving For 'd': A Step-by-Step Guide

Hey math enthusiasts! Today, we're diving into a classic algebra problem: solving for a variable. Specifically, we'll tackle the equation -61d - 5(61 - 11d) = 43. Don't worry if it looks a bit intimidating at first; we'll break it down into easy-to-follow steps. This guide will walk you through the process, ensuring you understand each move and, more importantly, why we make it. So, grab your pencils, and let's get started. Solving equations like this is a fundamental skill in algebra, and mastering it opens the door to more complex mathematical concepts. This problem involves simplifying expressions, applying the distributive property, and isolating the variable 'd'. It's a great exercise to build your algebra muscles! We'll begin by focusing on the left side of the equation, aiming to simplify it. Remember, the goal is always to get 'd' by itself on one side of the equation. Understanding this process builds a strong foundation for tackling more challenging algebraic problems. Get ready to flex those brain muscles; this is going to be fun.

Step 1: Distribute and Simplify

Okay guys, the first thing we're gonna do is address those parentheses. We've got -61d - 5(61 - 11d) = 43. To get rid of the parentheses, we'll use the distributive property. This means multiplying the '-5' by each term inside the parentheses. So, '-5' times '61' is '-305', and '-5' times '-11d' is '+55d'. Our equation now looks like this: -61d - 305 + 55d = 43. Notice how we've eliminated the parentheses and started simplifying the left side of the equation. This is a crucial step towards isolating 'd'.

Now, let's combine like terms on the left side. We have '-61d' and '+55d'. Combining these gives us '-6d'. Our equation is now -6d - 305 = 43. See how we're gradually simplifying the equation? Each step brings us closer to solving for 'd'. The distributive property is a fundamental concept in algebra, and understanding it is key to solving many types of equations. Mastering this skill makes complex equations much easier to handle. Remember, the goal is to make the equation simpler so that we can isolate 'd' easily. We're on the right track!

This simplification process makes the equation much more manageable. The distributive property allows us to eliminate parentheses, which often obscure the underlying structure of the equation. Combining like terms further streamlines the equation by reducing the number of terms we need to work with. These steps are essential for ensuring an accurate solution.

Step 2: Isolate the Variable Term

Alright, next up, we want to get the term with 'd' (which is '-6d') by itself on one side of the equation. To do this, we need to get rid of the '-305' that's hanging out on the left side. We can do this by adding 305 to both sides of the equation. This is a critical rule in algebra: whatever you do to one side of the equation, you must do to the other. So, we add 305 to both sides: -6d - 305 + 305 = 43 + 305.

This simplifies to -6d = 348. See how we're inching closer to getting 'd' by itself? We've successfully isolated the term with the variable on the left side. This step is about maintaining the balance of the equation. By adding 305 to both sides, we've ensured that the equation remains true. This is the bedrock of solving equations: keeping everything balanced. Think of an equation like a scale; to keep it balanced, any change on one side must be mirrored on the other.

By systematically isolating the variable term, we're making the equation easier to solve. This process is not just about performing operations but also about understanding the principles behind them. Each step we take brings us closer to the solution, and recognizing the logic behind each step is important. Remember, algebra is a game of logic, and each move should be deliberate and purposeful.

Step 3: Solve for 'd'

Almost there, everyone! Now that we have -6d = 348, the final step is to isolate 'd'. To do this, we need to get rid of the '-6' that's multiplying 'd'. We achieve this by dividing both sides of the equation by '-6'. So, we have -6d / -6 = 348 / -6. This gives us d = -58. And there you have it! We've solved for 'd'! The value of 'd' that satisfies the original equation is '-58'.

To ensure our answer is correct, we can substitute '-58' back into the original equation and see if it holds true. This is called verifying the solution. Plugging in '-58' for 'd' in -61d - 5(61 - 11d) = 43, we get -61(-58) - 5(61 - 11(-58)) = 43. Simplifying this gives us 3538 - 5(61 + 638) = 43, which further simplifies to 3538 - 5(699) = 43. Next, we have 3538 - 3495 = 43, and finally, 43 = 43. Since the equation holds true, our solution is correct. Always verifying your answer is a great habit to develop in mathematics.

By dividing both sides by '-6', we've effectively isolated 'd' and found its value. This step showcases the inverse relationship between multiplication and division. Each operation we perform is designed to undo the operations applied to the variable, ultimately revealing its value. This highlights the importance of understanding mathematical operations and their relationships to ensure correct solutions.

Conclusion: Practice Makes Perfect

So there you have it, guys! We successfully solved for 'd' in the equation -61d - 5(61 - 11d) = 43. We went through the steps of distributing, simplifying, isolating the variable term, and finally, solving for 'd'. Remember, solving for variables is a core concept in algebra, and understanding this process will greatly help you with more advanced math problems. The key to mastering algebra, like any other skill, is practice. So, don't stop here! Try solving similar equations on your own. Practice different types of problems to boost your skills and confidence.

If you found this guide helpful, feel free to share it with your friends or classmates. The more we learn together, the better we become! Keep practicing, keep learning, and don't be afraid to ask questions. Good luck, and happy solving!

This detailed walkthrough provides a solid understanding of how to solve for 'd'. By explaining each step and its rationale, we've made the process accessible and easy to understand. The inclusion of verification emphasizes the importance of accuracy. Remember, practice is essential. Apply these steps to other problems, and you'll become confident in your algebra skills. Keep up the excellent work!

This step-by-step approach not only solves the problem but also provides a clear explanation of the underlying concepts, reinforcing the importance of each step and helping build a stronger foundation in algebra. It emphasizes the importance of practice and offers encouragement to continue learning and exploring the world of mathematics.