Solving -d/48 = -11/4 A Step-by-Step Guide

In the realm of mathematics, solving linear equations is a fundamental skill. These equations, characterized by a variable raised to the power of one, are ubiquitous in various fields, from basic algebra to advanced calculus and real-world applications. Mastering the techniques to solve them is crucial for anyone seeking to excel in mathematics and related disciplines. This article delves into a step-by-step approach to solving the equation $-\frac{d}{48} = -\frac{11}{4}$, providing a detailed explanation of each step involved. By understanding the underlying principles and applying them systematically, you can confidently tackle similar equations and enhance your problem-solving abilities.

This equation, $-\frac{d}{48} = -\frac{11}{4}$, falls under the category of linear equations, where our goal is to isolate the variable 'd' and determine its value. The variable 'd' represents an unknown quantity, and our task is to manipulate the equation using algebraic operations to find the specific value that satisfies the equation. The equation involves fractions, which adds a layer of complexity, but with a clear understanding of the principles of fraction manipulation and equation solving, we can systematically arrive at the solution. Linear equations are the building blocks of more complex mathematical concepts, so a solid grasp of how to solve them is essential for further mathematical studies.

Let's embark on a journey to dissect and conquer the equation $-\frac{d}{48} = -\frac{11}{4}$. Our primary objective is to isolate the variable 'd' on one side of the equation. This means we need to undo any operations that are currently affecting 'd'. In this case, 'd' is being divided by 48 and multiplied by -1. To isolate 'd', we need to reverse these operations. The strategy we'll employ is to multiply both sides of the equation by -48. This will effectively cancel out the -48 in the denominator on the left side, leaving 'd' isolated. It's crucial to remember that any operation performed on one side of the equation must also be performed on the other side to maintain the equality. This principle of balance is fundamental to solving equations correctly. By meticulously applying this strategy, we can systematically unravel the equation and reveal the value of 'd'.

Step 1: Multiply Both Sides by -48

To isolate 'd', we need to eliminate the denominator and the negative sign associated with it. The most effective way to achieve this is by multiplying both sides of the equation by -48. This operation will cancel out the -48 on the left side, leaving 'd' by itself. On the right side, we'll have a multiplication of two negative fractions, which will result in a positive value. Remember, multiplying both sides of an equation by the same value maintains the equality, ensuring that we're performing a valid operation. This step is crucial in simplifying the equation and bringing us closer to the solution. Let's see how this unfolds:

$$-48 \times \left(-\frac{d}{48}\right) = -48 \times \left(-\frac{11}{4}\right)$$

Step 2: Simplify Both Sides

Now that we've multiplied both sides by -48, it's time to simplify the equation. On the left side, the -48 in the numerator and the -48 in the denominator cancel each other out, leaving us with simply 'd'. This is precisely what we aimed for – isolating the variable. On the right side, we have a multiplication of -48 and -11/4. To simplify this, we can first address the signs: a negative times a negative results in a positive. Then, we can perform the multiplication of the numbers. We can think of -48 as -48/1, making it easier to multiply with the fraction -11/4. This step involves basic arithmetic operations, but it's essential to perform them accurately to arrive at the correct solution. By carefully simplifying both sides, we transform the equation into a more manageable form.

$$d = \frac{48 \times 11}{4}$$

Step 3: Further Simplification

We're not quite at the final answer yet. The right side of the equation, $\frac{48 \times 11}{4}$, can be simplified further. Notice that 48 and 4 have a common factor of 4. We can divide both 48 and 4 by 4 to simplify the fraction before performing the multiplication. This technique of simplifying fractions before multiplying makes the calculation easier and reduces the chances of making errors. By dividing 48 by 4, we get 12. This leaves us with 12 multiplied by 11. Now, the multiplication is much simpler to perform. This step demonstrates the importance of looking for opportunities to simplify expressions before carrying out more complex operations.

$$d = 12 \times 11$$

Step 4: Calculate the Final Value of d

We've reached the final step in our journey to solve for 'd'. The equation has been simplified to $d = 12 \times 11$. Now, all that remains is to perform the multiplication. Multiplying 12 by 11 is a straightforward calculation that yields 132. This value represents the solution to the original equation. We have successfully isolated 'd' and determined its value through a series of logical and systematic steps. This final calculation brings our solution to fruition and demonstrates the power of algebraic manipulation in solving equations.

$$d = 132$$

Before we declare victory, it's always prudent to verify our solution. This crucial step involves substituting the value we found for 'd' back into the original equation and checking if it holds true. By substituting d = 132 into the equation $-\frac{d}{48} = -\frac{11}{4}$, we can confirm whether our solution satisfies the equality. This process not only validates our answer but also provides an opportunity to identify any potential errors in our calculations. If the equation holds true after the substitution, we can confidently conclude that our solution is correct. Verification is an essential practice in mathematics, ensuring the accuracy and reliability of our results.

Substituting d = 132 into the Original Equation

To verify our solution, we'll replace 'd' with 132 in the original equation: $-\frac{d}{48} = -\frac{11}{4}$. This substitution will allow us to check if both sides of the equation are equal when d = 132. This process is a cornerstone of problem-solving in mathematics, providing a tangible way to confirm the correctness of our answer. By meticulously performing this substitution, we can gain confidence in our solution and ensure that we've arrived at the accurate value for 'd'.

$$-\frac{132}{48} = -\frac{11}{4}$$

Simplifying the Left-Hand Side

Now that we've substituted d = 132 into the equation, we need to simplify the left-hand side: $-\frac{132}{48}$. Both 132 and 48 share common factors, which allows us to reduce the fraction to its simplest form. Identifying and canceling out these common factors is a crucial step in verifying our solution. By simplifying the fraction, we can more easily compare it to the right-hand side of the equation and determine if the equality holds true. This process highlights the importance of fraction manipulation in solving and verifying mathematical equations.

Both 132 and 48 are divisible by 12. Dividing 132 by 12 gives us 11, and dividing 48 by 12 gives us 4. Therefore, the simplified fraction is:

$$-\frac{11}{4} = -\frac{11}{4}$$

Conclusion of Verification

After simplifying the left-hand side of the equation, we arrive at $-\frac{11}{4}$, which is exactly the same as the right-hand side of the equation. This confirms that our solution, d = 132, is indeed correct. The verification process has successfully demonstrated that substituting d = 132 into the original equation satisfies the equality. This final step solidifies our understanding of the problem and the solution, reinforcing the importance of verification in mathematical problem-solving. We can confidently conclude that d = 132 is the accurate solution to the equation $-\frac{d}{48} = -\frac{11}{4}$.

In conclusion, we have successfully solved the equation $-\frac{d}{48} = -\frac{11}{4}$ by systematically isolating the variable 'd'. Our step-by-step approach involved multiplying both sides of the equation by -48, simplifying the resulting expression, and ultimately arriving at the solution d = 132. Furthermore, we rigorously verified our solution by substituting it back into the original equation, confirming its accuracy. This process highlights the importance of understanding and applying algebraic principles to solve linear equations. The ability to solve such equations is a fundamental skill in mathematics, with applications spanning various fields. By mastering these techniques, you can confidently tackle a wide range of mathematical problems and enhance your problem-solving abilities.

The key takeaways from this exercise include the importance of maintaining balance in equations by performing the same operations on both sides, the strategic use of multiplication and division to isolate variables, and the crucial role of verification in ensuring the accuracy of solutions. By consistently applying these principles, you can develop a strong foundation in algebra and excel in your mathematical endeavors. Remember, practice is key to mastering these skills, so continue to challenge yourself with different types of equations and problems. With dedication and perseverance, you can unlock the power of mathematics and apply it to solve real-world challenges.