Solving For N How To Solve The Equation (3n+3)/5 = (5n-1)/9
This article delves into the process of solving the equation (3n+3)/5 = (5n-1)/9 for the variable 'n'. We will explore the step-by-step method to arrive at the correct solution, ensuring a clear understanding of the underlying algebraic principles. This guide is designed for students, educators, and anyone looking to enhance their problem-solving skills in mathematics. By the end of this article, you will be equipped with the knowledge to tackle similar algebraic equations with confidence.
Understanding the Problem
The given equation is a linear equation involving fractions. To find the value of n, we need to eliminate the fractions and isolate the variable on one side of the equation. This involves a series of algebraic manipulations that preserve the equality. Let's break down the process into manageable steps.
1. Eliminate the Fractions
The first step in solving this equation is to eliminate the fractions. This can be achieved by multiplying both sides of the equation by the least common multiple (LCM) of the denominators, which are 5 and 9. The LCM of 5 and 9 is 45. So, we multiply both sides of the equation by 45:
45 * (3n + 3) / 5 = 45 * (5n - 1) / 9
This simplifies to:
9 * (3n + 3) = 5 * (5n - 1)
2. Distribute the Constants
Next, we distribute the constants on both sides of the equation. This means multiplying the numbers outside the parentheses with each term inside the parentheses:
9 * 3n + 9 * 3 = 5 * 5n - 5 * 1
Which simplifies to:
27n + 27 = 25n - 5
3. Isolate the Variable
Now, we need to isolate the variable n on one side of the equation. To do this, we can subtract 25**n** from both sides:
27n - 25n + 27 = 25n - 25n - 5
This simplifies to:
2n + 27 = -5
4. Isolate the Constant Term
Next, we isolate the constant term by subtracting 27 from both sides:
2n + 27 - 27 = -5 - 27
This simplifies to:
2n = -32
5. Solve for n
Finally, we solve for n by dividing both sides by 2:
2n / 2 = -32 / 2
This gives us the solution:
n = -16
Therefore, the value of n that makes the equation true is -16.
Verifying the Solution
To ensure our solution is correct, we can substitute n = -16 back into the original equation:
(3 * (-16) + 3) / 5 = (5 * (-16) - 1) / 9
Simplifying the left side:
(-48 + 3) / 5 = -45 / 5 = -9
Simplifying the right side:
(-80 - 1) / 9 = -81 / 9 = -9
Since both sides of the equation are equal (-9 = -9), our solution n = -16 is correct.
Common Mistakes to Avoid
When solving equations like this, there are several common mistakes that students often make. Being aware of these pitfalls can help you avoid them.
1. Incorrectly Distributing Constants
A frequent error is not properly distributing constants across terms within parentheses. For example, in the step 9 * (3n + 3) = 5 * (5n - 1), it's crucial to multiply 9 by both 3**n** and 3, and similarly, 5 by both 5**n** and -1. Failing to do so can lead to an incorrect equation and, consequently, a wrong solution.
2. Arithmetic Errors
Simple arithmetic mistakes, such as adding or subtracting numbers incorrectly, can derail the entire solution process. Double-checking each step, especially when dealing with negative numbers, can prevent these errors.
3. Not Applying Operations to Both Sides
The fundamental principle of solving equations is to maintain balance by performing the same operation on both sides. For instance, if you subtract a number from one side, you must subtract the same number from the other side. Neglecting this principle will lead to an incorrect solution.
4. Forgetting to Verify the Solution
It’s always a good practice to substitute the solution back into the original equation to verify its correctness. This step helps catch any mistakes made during the solving process and ensures the final answer is accurate.
5. Misunderstanding Order of Operations
Adhering to the correct order of operations (PEMDAS/BODMAS) is crucial. Perform operations in the correct sequence: Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), Addition and Subtraction (from left to right). Skipping or misinterpreting this order can result in an incorrect solution.
Tips for Solving Algebraic Equations
To improve your skills in solving algebraic equations, consider the following tips:
1. Practice Regularly
Like any mathematical skill, solving equations requires practice. The more you practice, the more comfortable and proficient you will become.
2. Break Down Complex Problems
When faced with a complex equation, break it down into smaller, more manageable steps. This makes the problem less intimidating and easier to solve.
3. Show Your Work
Write down each step of your solution process. This helps you keep track of your progress and makes it easier to identify any mistakes.
4. Use Estimation to Check Your Answer
Before solving an equation, estimate the answer. This can help you determine if your final solution is reasonable.
5. Understand the Underlying Principles
Focus on understanding the underlying principles of algebra, rather than just memorizing steps. This will enable you to solve a wider range of problems.
Conclusion
Solving the equation (3**n+3)/5 = (5n**-1)/9 involves a series of algebraic steps, including eliminating fractions, distributing constants, isolating the variable, and solving for n. By following these steps carefully and avoiding common mistakes, we arrive at the correct solution, which is n = -16. Verifying the solution by substituting it back into the original equation confirms its accuracy. Consistent practice and a solid understanding of algebraic principles are key to mastering equation-solving skills. This comprehensive guide aims to provide you with the tools and knowledge necessary to confidently tackle similar problems in the future. Remember to break down complex problems, show your work, and always verify your solution to ensure accuracy.
Practice Problems
To further solidify your understanding, try solving these practice problems:
- (2**n** + 5) / 3 = (4**n** - 1) / 2
- (n + 7) / 4 = (3**n** - 2) / 5
- (5**n** - 3) / 2 = (2**n** + 1) / 3
Work through these problems using the steps outlined in this guide, and remember to verify your solutions. Good luck!
By mastering the techniques discussed in this article, you'll be well-prepared to solve a wide range of algebraic equations and enhance your mathematical proficiency. Keep practicing, and don't hesitate to review the steps whenever needed. With dedication and a clear understanding of the principles involved, you can conquer any equation that comes your way.