Solve The Equation:$\[ \frac{4}{3} - \frac{7}{n} = \frac{1}{6} \\]

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Introduction

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In this article, we will delve into the world of mathematics and explore a simple yet intriguing equation. The equation in question is $\frac{4}{3} - \frac{7}{n} = \frac{1}{6}$, where $n$ is an unknown variable. Our goal is to solve for $n$ and understand the underlying mathematical concepts that make this equation tick.

Understanding the Equation

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Before we dive into the solution, let's take a closer look at the equation and break it down into its constituent parts. We have three fractions: $\frac{4}{3}$, $\frac{7}{n}$, and $\frac{1}{6}$. The equation states that the difference between the first two fractions is equal to the third fraction.

The Role of Fractions

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Fractions are a fundamental concept in mathematics, representing a part of a whole. In this equation, the fractions are used to represent different values and relationships between them. Understanding fractions is crucial to solving this equation.

The Unknown Variable

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The variable $n$ is the unknown quantity that we need to solve for. It is the key to unlocking the solution to this equation. As we progress through the solution, we will see how $n$ is used to balance the equation.

Solving the Equation

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Now that we have a good understanding of the equation and its components, let's dive into the solution. We will use a step-by-step approach to solve for $n$.

Step 1: Multiply Both Sides by $n$

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To eliminate the fraction $\frac{7}{n}$, we can multiply both sides of the equation by $n$. This will give us:

$$4n - 7 = \frac{n}{6}$$

Step 2: Multiply Both Sides by 6

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To get rid of the fraction $\frac{n}{6}$, we can multiply both sides of the equation by 6. This will give us:

$$24n - 42 = n$$

Step 3: Subtract $n$ from Both Sides

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To isolate the term $n$, we can subtract $n$ from both sides of the equation. This will give us:

$$23n - 42 = 0$$

Step 4: Add 42 to Both Sides

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To get rid of the constant term -42, we can add 42 to both sides of the equation. This will give us:

$$23n = 42$$

Step 5: Divide Both Sides by 23

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To solve for $n$, we can divide both sides of the equation by 23. This will give us:

$$n = \frac{42}{23}$$

Conclusion

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And there you have it! We have successfully solved the equation $\frac{4}{3} - \frac{7}{n} = \frac{1}{6}$ and found the value of $n$. The solution involved a series of steps, including multiplying both sides by $n$, multiplying both sides by 6, subtracting $n$ from both sides, adding 42 to both sides, and finally dividing both sides by 23.

Final Answer

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The final answer to the equation is:

$$n = \frac{42}{23}$$

Applications of the Solution

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The solution to this equation has various applications in mathematics and real-world scenarios. For example, it can be used to solve systems of linear equations, model real-world problems, and even optimize functions.

Tips and Tricks

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When solving equations involving fractions, it's essential to remember the following tips and tricks:

• Multiply both sides of the equation by the least common multiple (LCM) of the denominators to eliminate fractions.
• Use algebraic manipulations, such as adding or subtracting the same value to both sides, to isolate the variable.
• Be careful when multiplying or dividing both sides of the equation by a fraction, as this can lead to incorrect solutions.

Common Mistakes

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When solving equations involving fractions, it's easy to make mistakes. Here are some common pitfalls to avoid:

• Failing to multiply both sides of the equation by the LCM of the denominators.
• Not using algebraic manipulations to isolate the variable.
• Making errors when multiplying or dividing both sides of the equation by a fraction.

Real-World Examples

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The solution to this equation has various real-world applications. For example:

• In finance, the equation can be used to calculate the interest rate on a loan.
• In physics, the equation can be used to model the motion of an object under the influence of gravity.
• In engineering, the equation can be used to design and optimize systems.

Conclusion

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In conclusion, solving the equation $\frac{4}{3} - \frac{7}{n} = \frac{1}{6}$ requires a step-by-step approach and a solid understanding of fractions and algebraic manipulations. By following the solution outlined in this article, you can successfully solve for $n$ and apply the solution to real-world scenarios.

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Introduction

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In our previous article, we explored the equation $\frac{4}{3} - \frac{7}{n} = \frac{1}{6}$ and provided a step-by-step solution to find the value of $n$. In this article, we will address some of the most frequently asked questions (FAQs) related to this equation and provide additional insights to help you better understand the solution.

Q&A

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Q: What is the final answer to the equation?

A: The final answer to the equation is $n = \frac{42}{23}$.

Q: How do I know which steps to take when solving the equation?

A: When solving the equation, it's essential to follow a step-by-step approach. Start by multiplying both sides of the equation by $n$ to eliminate the fraction $\frac{7}{n}$. Then, multiply both sides by 6 to get rid of the fraction $\frac{n}{6}$. Finally, subtract $n$ from both sides, add 42 to both sides, and divide both sides by 23 to solve for $n$.

Q: What if I make a mistake when solving the equation?

A: If you make a mistake when solving the equation, don't worry! Simply re-evaluate your steps and correct any errors. Remember to check your work by plugging the solution back into the original equation.

Q: Can I use this equation to solve other problems?

A: Yes, the equation $\frac{4}{3} - \frac{7}{n} = \frac{1}{6}$ can be used to solve other problems. For example, you can use it to model real-world scenarios, such as calculating interest rates or designing systems.

Q: What are some common mistakes to avoid when solving the equation?

A: Some common mistakes to avoid when solving the equation include:

• Failing to multiply both sides of the equation by the least common multiple (LCM) of the denominators.
• Not using algebraic manipulations to isolate the variable.
• Making errors when multiplying or dividing both sides of the equation by a fraction.

Q: How can I apply the solution to real-world scenarios?

A: The solution to the equation can be applied to various real-world scenarios, such as:

• Calculating interest rates on loans
• Modeling the motion of objects under the influence of gravity
• Designing and optimizing systems

Q: What are some additional tips and tricks for solving equations involving fractions?

A: Some additional tips and tricks for solving equations involving fractions include:

• Using algebraic manipulations to isolate the variable
• Checking your work by plugging the solution back into the original equation
• Being careful when multiplying or dividing both sides of the equation by a fraction

Conclusion

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In conclusion, solving the equation $\frac{4}{3} - \frac{7}{n} = \frac{1}{6}$ requires a step-by-step approach and a solid understanding of fractions and algebraic manipulations. By following the solution outlined in this article and addressing some of the most frequently asked questions, you can successfully solve for $n$ and apply the solution to real-world scenarios.

Final Thoughts

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Solving equations involving fractions can be challenging, but with practice and patience, you can master the skills needed to tackle even the most complex problems. Remember to stay focused, check your work, and be careful when multiplying or dividing both sides of the equation by a fraction. With these tips and tricks, you'll be well on your way to becoming a math whiz!

Additional Resources

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For more information on solving equations involving fractions, check out the following resources:

• Khan Academy: Solving Equations with Fractions
• Mathway: Solving Equations with Fractions
• Wolfram Alpha: Solving Equations with Fractions

Conclusion

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In conclusion, solving the equation $\frac{4}{3} - \frac{7}{n} = \frac{1}{6}$ requires a step-by-step approach and a solid understanding of fractions and algebraic manipulations. By following the solution outlined in this article and addressing some of the most frequently asked questions, you can successfully solve for $n$ and apply the solution to real-world scenarios.