Solving for d in the equation -10d - 12 - 19 = -d + 14 involves isolating the variable 'd' on one side of the equation. This requires a systematic approach, utilizing algebraic principles to simplify and rearrange the equation until 'd' is by itself. This process includes combining like terms, moving terms across the equals sign, and ultimately dividing to find the value of 'd'. Mastering this technique is vital for tackling various mathematical problems, especially in algebra and calculus, making it a fundamental skill for students and professionals alike. Understanding the underlying concepts enhances your mathematical toolbox and prepares you for more complex problem-solving scenarios.
The initial equation presented is -10d - 12 - 19 = -d + 14. The first step in solving for 'd' is to simplify the equation by combining like terms on each side. On the left-hand side, we have two constant terms, -12 and -19, which can be combined. Adding these together gives us -12 - 19 = -31. So, the equation now becomes -10d - 31 = -d + 14. This simplification makes the equation easier to work with and reduces the chance of errors in subsequent steps. By combining like terms, we reduce the complexity of the equation and bring it closer to a form where the variable 'd' can be isolated, demonstrating the foundational principle of simplification in algebraic manipulations. Simplifying equations not only makes them more manageable but also aids in visualizing the relationships between variables and constants more clearly.
After simplifying the equation to -10d - 31 = -d + 14, the next key step is to rearrange the terms so that all terms involving 'd' are on one side of the equation and the constant terms are on the other side. This is achieved by using inverse operations. To move the '-d' term from the right side to the left side, we add 'd' to both sides of the equation. This gives us -10d + d - 31 = -d + d + 14, which simplifies to -9d - 31 = 14. Similarly, to move the constant term '-31' from the left side to the right side, we add 31 to both sides of the equation. This results in -9d - 31 + 31 = 14 + 31, which simplifies to -9d = 45. The principle of adding the same value to both sides ensures that the equation remains balanced, preserving the equality. This rearrangement is a critical step in isolating the variable, allowing us to proceed towards solving for 'd'.
With the equation rearranged to -9d = 45, the next step is to isolate 'd' by undoing the multiplication. Since 'd' is being multiplied by -9, we perform the inverse operation, which is division. We divide both sides of the equation by -9 to maintain the balance. This gives us (-9d) / -9 = 45 / -9. On the left side, the -9 in the numerator and the -9 in the denominator cancel each other out, leaving just 'd'. On the right side, 45 divided by -9 is -5. Therefore, the equation simplifies to d = -5. This division step is crucial in solving for the variable, as it separates 'd' from its coefficient, revealing its value. This process highlights the importance of inverse operations in algebra, where each operation is undone by its inverse to isolate the variable of interest.
To ensure the accuracy of our solution, it is essential to verify the solution. This is done by substituting the value we found for 'd', which is -5, back into the original equation: -10d - 12 - 19 = -d + 14. Substituting d = -5, we get -10(-5) - 12 - 19 = -(-5) + 14. Now, we simplify both sides of the equation. On the left side, -10 multiplied by -5 is 50, so we have 50 - 12 - 19. Subtracting 12 from 50 gives us 38, and then subtracting 19 gives us 19. So, the left side simplifies to 19. On the right side, the negative of -5 is 5, so we have 5 + 14, which equals 19. Therefore, the right side also simplifies to 19. Since both sides of the equation equal 19, our solution d = -5 is verified to be correct. Verification is a critical step in problem-solving as it confirms the accuracy of the solution and helps to identify any potential errors made during the solving process. This practice reinforces the understanding of algebraic principles and ensures confidence in the final answer.
While the step-by-step method described above is a standard approach, there are alternative methods for solving linear equations that can sometimes be more efficient or easier to understand, depending on the individual's preference and the complexity of the equation. One such method involves a slightly different order of operations. Instead of combining constants first, you might choose to move all terms involving 'd' to one side and all constant terms to the other side right away. For example, in the equation -10d - 12 - 19 = -d + 14, you could add 10d to both sides and subtract 14 from both sides in the first step. This would give you -12 - 19 - 14 = -d + 10d, which simplifies to -45 = 9d. Then, dividing both sides by 9, you would get d = -5. This approach can be beneficial when dealing with more complex equations where combining like terms first might not be as straightforward. Another approach involves using the properties of equality in a slightly different way, such as multiplying or dividing both sides of the equation by a constant to clear fractions or decimals early in the process. The key is to understand the fundamental principles of algebraic manipulation and apply them in a way that makes the solving process as clear and efficient as possible. Exploring these alternative methods not only enhances problem-solving skills but also deepens the understanding of the underlying mathematical concepts.
When solving linear equations, it's easy to make mistakes if you're not careful. One common mistake is forgetting to apply an operation to all terms in the equation. For example, if you have the equation -10d - 31 = -d + 14 and you add 'd' to both sides, you need to make sure you add 'd' to both the -10d term and the -31 term on the left side, and to both the -d term and the 14 term on the right side. Another frequent error is making mistakes with signs, especially when dealing with negative numbers. For instance, when moving a term from one side of the equation to the other, remember to change its sign. If you have -9d = 45, you correctly divide both sides by -9 to get d = -5, but it's easy to forget the negative sign and end up with d = 5. To avoid these mistakes, it's helpful to write out each step clearly and carefully, double-checking your work as you go. It's also a good idea to substitute your solution back into the original equation to make sure it works. If the two sides of the equation don't match after you substitute your solution, you know you've made a mistake somewhere and need to go back and check your work. Paying close attention to detail and systematically checking each step will significantly reduce the likelihood of errors. Additionally, practicing a variety of equations helps in recognizing patterns and avoiding common pitfalls.
Solving for variables like 'd' is not just an abstract mathematical exercise; it has numerous real-world applications. Linear equations, the type of equation we've been working with, are used to model a wide range of phenomena in science, engineering, economics, and everyday life. For instance, in physics, you might use a linear equation to calculate the distance traveled by an object moving at a constant speed over a certain amount of time. If you know the speed and the time, you can solve for the distance. Similarly, in finance, you might use a linear equation to calculate the simple interest earned on an investment. If you know the principal amount, the interest rate, and the time period, you can solve for the interest earned. In business, linear equations can be used to model the relationship between costs, revenue, and profit. By setting up an equation that represents the profit as a function of the number of units sold, you can solve for the number of units you need to sell to break even or achieve a certain profit target. Even in everyday situations, solving for variables can be useful. For example, if you're planning a road trip and you know the distance you need to travel and the average speed you'll be driving, you can solve for the time it will take to reach your destination. The ability to solve for variables is a powerful tool that enables us to understand and make predictions about the world around us. This skill is invaluable in both professional settings and personal decision-making, highlighting the practical significance of mastering algebraic techniques.
In conclusion, solving for 'd' in the equation -10d - 12 - 19 = -d + 14 involves a systematic application of algebraic principles. By combining like terms, rearranging the equation, and using inverse operations, we successfully isolated 'd' and found its value to be -5. Verifying this solution by substituting it back into the original equation confirmed its accuracy. Throughout this process, we've emphasized the importance of each step, highlighting common mistakes to avoid and exploring alternative methods for solving linear equations. Moreover, we've demonstrated the real-world applications of solving for variables, illustrating its relevance in various fields, from science and engineering to finance and everyday life. The ability to solve for variables is a fundamental skill in mathematics and beyond. It empowers us to analyze problems, make informed decisions, and understand the relationships between different quantities. Mastering this skill opens doors to more advanced mathematical concepts and enhances our problem-solving capabilities in a wide range of contexts. By understanding the underlying principles and practicing consistently, we can confidently tackle more complex equations and apply these skills to real-world challenges.