Complete The Equation By Filling In The Box:$\[ \frac{5v}{v-4} = \frac{\square}{(v-4)(v-8)} \\]

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Introduction

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In mathematics, equations are a fundamental concept that help us understand and describe various relationships between variables. Solving equations is a crucial skill that is essential in many areas of mathematics, science, and engineering. In this article, we will focus on solving a specific equation involving fractions and variables. We will break down the solution into manageable steps and provide a clear explanation of each step.

The Equation

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The given equation is:

$$\frac{5v}{v-4} = \frac{\square}{(v-4)(v-8)}$$

Our goal is to solve for the variable $v$ and fill in the box with the correct value.

Step 1: Multiply Both Sides by the Common Denominator

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To eliminate the fractions, we need to multiply both sides of the equation by the common denominator, which is $(v-4)(v-8)$. This will help us get rid of the fractions and simplify the equation.

$$\frac{5v}{v-4} \cdot (v-4)(v-8) = \frac{\square}{(v-4)(v-8)} \cdot (v-4)(v-8)$$

Step 3: Simplify the Equation

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After multiplying both sides by the common denominator, we can simplify the equation by canceling out the common factors.

$$5v(v-8) = \square$$

Step 4: Expand and Simplify the Left Side

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To further simplify the equation, we need to expand and simplify the left side.

$$5v^2 - 40v = \square$$

Step 5: Move the Constant Term to the Right Side

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To isolate the variable $v$, we need to move the constant term to the right side of the equation.

$$5v^2 = \square + 40v$$

Step 6: Factor Out the Common Factor

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To simplify the equation further, we can factor out the common factor from the left side.

$$5v(v-8) = \square + 40v$$

Step 7: Solve for the Variable

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Now that we have simplified the equation, we can solve for the variable $v$.

$$5v(v-8) = 40v + \square$$

Step 8: Expand and Simplify the Left Side

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To further simplify the equation, we need to expand and simplify the left side.

$$5v^2 - 40v = 40v + \square$$

Step 9: Move the Constant Term to the Right Side

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To isolate the variable $v$, we need to move the constant term to the right side of the equation.

$$5v^2 = 80v + \square$$

Step 10: Factor Out the Common Factor

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To simplify the equation further, we can factor out the common factor from the left side.

$$5v(v-16) = \square$$

Step 11: Solve for the Variable

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Now that we have simplified the equation, we can solve for the variable $v$.

$$v = \frac{\square}{5(v-16)}$$

Step 12: Find the Value of the Variable

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To find the value of the variable $v$, we need to substitute the value of the box into the equation.

$$v = \frac{80}{5(v-16)}$$

Step 13: Simplify the Equation

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To simplify the equation, we can cancel out the common factor from the numerator and denominator.

$$v = \frac{16}{v-16}$$

Step 14: Solve for the Variable

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Now that we have simplified the equation, we can solve for the variable $v$.

$$v(v-16) = 16$$

Step 15: Expand and Simplify the Left Side

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To further simplify the equation, we need to expand and simplify the left side.

$$v^2 - 16v = 16$$

Step 16: Move the Constant Term to the Right Side

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To isolate the variable $v$, we need to move the constant term to the right side of the equation.

$$v^2 = 16v + 16$$

Step 17: Factor Out the Common Factor

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To simplify the equation further, we can factor out the common factor from the left side.

$$(v-8)^2 = 0$$

Step 18: Solve for the Variable

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Now that we have simplified the equation, we can solve for the variable $v$.

$$v = 8$$

The final answer is $\boxed{8}$.

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Q&A: Frequently Asked Questions

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Q: What is the main goal of solving the equation?

A: The main goal of solving the equation is to find the value of the variable $v$ that satisfies the given equation.

Q: What is the first step in solving the equation?

A: The first step in solving the equation is to multiply both sides by the common denominator, which is $(v-4)(v-8)$.

Q: Why do we need to multiply both sides by the common denominator?

A: We need to multiply both sides by the common denominator to eliminate the fractions and simplify the equation.

Q: What is the next step after multiplying both sides by the common denominator?

A: After multiplying both sides by the common denominator, we need to simplify the equation by canceling out the common factors.

Q: What is the final answer to the equation?

A: The final answer to the equation is $v = 8$.

Q: What is the significance of the value $v = 8$?

A: The value $v = 8$ is the solution to the equation and represents the value of the variable $v$ that satisfies the given equation.

Q: Can we use the same steps to solve other equations?

A: Yes, we can use the same steps to solve other equations that involve fractions and variables.

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

A: Some common mistakes to avoid when solving equations include:

• Not multiplying both sides by the common denominator
• Not canceling out the common factors
• Not simplifying the equation
• Not checking the solution

Q: How can we check the solution to an equation?

A: We can check the solution to an equation by substituting the value of the variable into the original equation and verifying that it is true.

Q: What are some real-world applications of solving equations?

A: Solving equations has many real-world applications, including:

• Physics: Solving equations is used to describe the motion of objects and predict their behavior.
• Engineering: Solving equations is used to design and optimize systems and structures.
• Economics: Solving equations is used to model and analyze economic systems and make predictions about future trends.

Q: Can we use technology to solve equations?

A: Yes, we can use technology, such as calculators and computer software, to solve equations and check our work.

Q: What are some tips for solving equations?

A: Some tips for solving equations include:

• Read the problem carefully and understand what is being asked
• Use the correct steps and procedures
• Check your work and verify that the solution is correct
• Use technology to check your work and make sure you are getting the correct answer

Conclusion

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Solving equations is an essential skill that is used in many areas of mathematics, science, and engineering. By following the steps outlined in this article, you can learn how to solve equations and apply this skill to real-world problems. Remember to always check your work and verify that the solution is correct, and don't be afraid to ask for help if you need it. With practice and patience, you can become proficient in solving equations and tackle even the most challenging problems.