Solving For D In The Equation D/3 + 2 = 9 A Step-by-Step Guide
In this article, we will explore how to solve a simple algebraic equation to find the value of the variable d. The equation we'll be working with is: d/3 + 2 = 9. This type of problem is fundamental in algebra and understanding how to solve it is crucial for more advanced mathematical concepts. We will break down the steps involved, providing a clear and comprehensive explanation for each. Whether you're a student learning algebra for the first time or someone looking to refresh your skills, this guide will walk you through the process step-by-step.
Understanding the Equation
Before we dive into solving the equation, it's important to understand what it represents. The equation d/3 + 2 = 9 is an algebraic statement that says: "When a number d is divided by 3, and then 2 is added to the result, the final answer is 9." Our goal is to find the specific value of d that makes this statement true. To achieve this, we will use the principles of algebraic manipulation, which involve performing the same operations on both sides of the equation to isolate the variable d. This ensures that the equation remains balanced and the solution we find is accurate. Understanding the equation in this way helps us approach the problem with a clear strategy and avoid common mistakes.
Step 1: Isolate the Term with 'd'
The first step in solving for d is to isolate the term that contains d, which in this case is d/3. To do this, we need to eliminate the constant term (+2) on the left side of the equation. We can achieve this by subtracting 2 from both sides of the equation. This is a crucial step in maintaining the balance of the equation. Subtracting 2 from both sides gives us:
d/3 + 2 - 2 = 9 - 2
This simplifies to:
d/3 = 7
By subtracting 2 from both sides, we have successfully isolated the term with d on one side of the equation, making it easier to proceed with the next step in solving for d. This step is a fundamental application of the properties of equality in algebra.
Step 2: Solve for 'd'
Now that we have isolated the term d/3, the next step is to solve for d itself. The equation we are working with is: d/3 = 7. To get d by itself, we need to undo the division by 3. We can do this by multiplying both sides of the equation by 3. This is the inverse operation of division and will effectively cancel out the division by 3 on the left side of the equation. Multiplying both sides by 3 gives us:
3 * (d/3) = 3 * 7
This simplifies to:
d = 21
Therefore, the value of d that satisfies the equation d/3 + 2 = 9 is 21. This step completes the process of solving for d and provides the solution to the equation. It demonstrates the importance of using inverse operations to isolate the variable we are trying to find.
Step 3: Verification of the Solution
To ensure that our solution d = 21 is correct, it's always a good practice to verify it by substituting it back into the original equation. This process helps to catch any potential errors made during the solving process. The original equation is: d/3 + 2 = 9. Now, let's substitute d with 21:
21/3 + 2 = 9
First, we perform the division:
7 + 2 = 9
Then, we add the numbers on the left side:
9 = 9
Since the equation holds true, our solution d = 21 is correct. This verification step reinforces the accuracy of our solution and provides confidence in our problem-solving process. It is a valuable habit to develop when working with algebraic equations.
Alternative method
Let's explore an alternative method for solving the equation d/3 + 2 = 9. This approach offers a slightly different perspective and can be helpful for those who prefer a different sequence of steps.
Step 1: Eliminate the Fraction
The alternative method begins by eliminating the fraction in the equation right away. This can simplify the equation and make it easier to work with for some people. To eliminate the fraction, we multiply every term in the equation by the denominator, which in this case is 3. This ensures that we maintain the balance of the equation while getting rid of the fraction. Multiplying both sides of the equation d/3 + 2 = 9 by 3 gives us:
3 * (d/3) + 3 * 2 = 3 * 9
Step 2: Simplify the Equation
Next, we simplify the equation by performing the multiplication operations. This step will eliminate the fraction and result in a more straightforward equation. Simplifying the equation from the previous step, we get:
d + 6 = 27
Step 3: Isolate 'd'
Now that we have a simplified equation without fractions, we need to isolate d. To do this, we subtract 6 from both sides of the equation. This will remove the constant term on the left side and bring us closer to finding the value of d. Subtracting 6 from both sides gives us:
d + 6 - 6 = 27 - 6
Step 4: Solve for 'd'
Finally, we solve for d by simplifying the equation. This will give us the value of d that satisfies the original equation. Simplifying the equation from the previous step, we get:
d = 21
Thus, using this alternative method, we also find that the value of d is 21. This demonstrates that there can be multiple valid approaches to solving the same algebraic equation.
Conclusion
In conclusion, we have successfully found the value of d in the equation d/3 + 2 = 9 using two different methods. Both methods led us to the solution d = 21. The first method involved isolating the term with d first and then solving for d, while the alternative method involved eliminating the fraction first. Understanding these different approaches can enhance your problem-solving skills and give you the flexibility to choose the method that you find most comfortable and efficient. Remember to always verify your solution by substituting it back into the original equation to ensure accuracy. Mastering these fundamental algebraic techniques is essential for success in mathematics and related fields. This step-by-step guide should provide a solid foundation for tackling similar algebraic problems in the future. Keep practicing and exploring different methods to strengthen your understanding and skills in algebra.
Frequently Asked Questions (FAQs)
To further solidify your understanding of solving algebraic equations like d/3 + 2 = 9, let's address some frequently asked questions.
1. What is the basic principle behind solving algebraic equations?
The fundamental principle behind solving algebraic equations is to isolate the variable you are trying to find on one side of the equation. This is achieved by performing the same operations on both sides of the equation to maintain equality. The goal is to undo the operations that are being applied to the variable until the variable is by itself. This often involves using inverse operations, such as addition to undo subtraction, multiplication to undo division, and vice versa.
2. Why is it important to perform the same operation on both sides of the equation?
Performing the same operation on both sides of the equation is crucial because it maintains the balance and equality of the equation. An equation is like a balanced scale; if you add or subtract something from one side, you must do the same to the other side to keep it balanced. Similarly, if you multiply or divide one side, you must do the same to the other. This ensures that the solution you find is valid and satisfies the original equation. If you only perform an operation on one side, you are essentially changing the equation and will likely arrive at an incorrect solution.
3. How do I know which operation to perform first when solving an equation?
The order of operations often used in reverse when solving equations, which is sometimes remembered by the acronym SADMEP (Subtraction, Addition, Division, Multiplication, Exponents, Parentheses). However, the specific order can depend on the equation. A general guideline is to first undo any addition or subtraction, then undo multiplication or division, and finally, deal with any exponents or parentheses. It's important to look at the equation and identify the operations that are being applied to the variable. Then, perform the inverse operations in the reverse order to isolate the variable. Practice and familiarity with different types of equations will help you develop an intuition for the best order of operations.
4. What is the purpose of verifying the solution?
Verifying the solution is a crucial step in the problem-solving process because it ensures that the solution you have found is correct. By substituting the solution back into the original equation, you can check if the equation holds true. This process helps to catch any potential errors made during the solving process, such as arithmetic mistakes or incorrect application of operations. Verifying the solution provides confidence in your answer and helps to avoid submitting incorrect solutions. It is a good habit to develop when working with any type of mathematical problem, not just algebraic equations.
5. Are there always multiple methods to solve an algebraic equation?
While not every algebraic equation has multiple distinct methods of solution, many equations can be solved using different approaches. The choice of method often depends on personal preference, the specific structure of the equation, and the solver's familiarity with different techniques. Understanding multiple methods can be beneficial because it provides flexibility in problem-solving and can sometimes lead to a more efficient solution. Exploring different approaches also deepens your understanding of the underlying mathematical concepts and principles. In the example of d/3 + 2 = 9, we saw two methods: one that isolated the term with d first and another that eliminated the fraction first. Both methods are valid and lead to the same correct solution.
