Solve For \[$ V \$\]:$\[ 1 = \frac{v+2}{v-4} + \frac{7v-42}{v-4} \\]

$$$$

Introduction

In this article, we will delve into solving a complex equation involving fractions. The given equation is $1 = \frac{v+2}{v-4} + \frac{7v-42}{v-4}$. Our goal is to isolate the variable $v$ and find its value. We will use algebraic techniques to simplify the equation and solve for $v$.

Understanding the Equation

The given equation is a linear equation involving two fractions. The first fraction is $\frac{v+2}{v-4}$, and the second fraction is $\frac{7v-42}{v-4}$. The equation states that the sum of these two fractions is equal to 1.

Simplifying the Equation

To simplify the equation, we can start by combining the two fractions on the right-hand side. Since both fractions have the same denominator, $v-4$, we can add them together.

$\frac{v+2}{v-4} + \frac{7v-42}{v-4} = \frac{(v+2) + (7v-42)}{v-4}$

Expanding the Numerator

Now, we can expand the numerator by combining like terms.

$\frac{(v+2) + (7v-42)}{v-4} = \frac{8v-40}{v-4}$

Simplifying the Fraction

We can simplify the fraction by dividing the numerator and denominator by their greatest common divisor, which is 4.

$\frac{8v-40}{v-4} = \frac{2(4v-20)}{v-4}$

Canceling Out the Common Factor

Now, we can cancel out the common factor of $v-4$ in the numerator and denominator.

$\frac{2(4v-20)}{v-4} = 2 \cdot \frac{4v-20}{v-4}$

Simplifying the Expression

We can simplify the expression by canceling out the common factor of $v-4$ in the numerator and denominator.

$2 \cdot \frac{4v-20}{v-4} = 2 \cdot \frac{4(v-5)}{v-4}$

Canceling Out the Common Factor

Now, we can cancel out the common factor of $v-4$ in the numerator and denominator.

$2 \cdot \frac{4(v-5)}{v-4} = 2 \cdot 4$

Simplifying the Expression

We can simplify the expression by multiplying the numbers.

$2 \cdot 4 = 8$

Solving for $v$

Now that we have simplified the equation, we can solve for $v$. We know that the equation is equal to 1, so we can set up the equation as follows:

$8 = 1$

Subtracting 1 from Both Sides

To isolate the variable $v$, we can subtract 1 from both sides of the equation.

$8 - 1 = 1 - 1$

Simplifying the Equation

We can simplify the equation by evaluating the expressions.

$7 = 0$

Dividing Both Sides by 7

To solve for $v$, we can divide both sides of the equation by 7.

$\frac{7}{7} = \frac{0}{7}$

Simplifying the Equation

We can simplify the equation by evaluating the expressions.

$1 = 0$

Conclusion

In this article, we solved the equation $1 = \frac{v+2}{v-4} + \frac{7v-42}{v-4}$ for the variable $v$. We used algebraic techniques to simplify the equation and isolate the variable $v$. However, we found that the equation is inconsistent, and there is no value of $v$ that satisfies the equation.

Final Answer

The final answer is $\boxed{No solution}$.

Discussion

The equation $1 = \frac{v+2}{v-4} + \frac{7v-42}{v-4}$ is an example of an inconsistent equation, which means that there is no value of $v$ that satisfies the equation. This can occur when the equation is contradictory, such as when the left-hand side and right-hand side are equal to different values.

Related Topics

• Solving linear equations
• Simplifying fractions
• Algebraic techniques

References

• [1] "Algebra" by Michael Artin
• [2] "Linear Algebra" by Jim Hefferon
• [3] "Solving Equations" by Paul Dawkins
  

$$

Introduction

In our previous article, we solved the equation $1 = \frac{v+2}{v-4} + \frac{7v-42}{v-4}$ and found that it is inconsistent, meaning that there is no value of $v$ that satisfies the equation. In this article, we will answer some common questions related to solving this equation.

Q: What is the main issue with the equation $1 = \frac{v+2}{v-4} + \frac{7v-42}{v-4}$?

A: The main issue with the equation is that the left-hand side and right-hand side are equal to different values. Specifically, the left-hand side is equal to 1, while the right-hand side is equal to 8.

Q: Why did we get a different value for the right-hand side?

A: We got a different value for the right-hand side because we simplified the equation incorrectly. When we combined the two fractions, we forgot to cancel out the common factor of $v-4$ in the numerator and denominator.

Q: What is the correct way to simplify the equation?

A: The correct way to simplify the equation is to combine the two fractions and then cancel out the common factor of $v-4$ in the numerator and denominator.

Q: Why is it important to check for inconsistencies in an equation?

A: It is important to check for inconsistencies in an equation because it can help us avoid making mistakes and ensure that our solutions are correct.

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

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

• Not checking for inconsistencies
• Not canceling out common factors
• Not simplifying the equation correctly
• Not following the order of operations

Q: How can I practice solving equations?

A: You can practice solving equations by working on problems and exercises in your textbook or online resources. You can also try solving equations on your own and then checking your solutions with a calculator or online tool.

Q: What are some resources for learning more about solving equations?

A: Some resources for learning more about solving equations include:

• Textbooks on algebra and mathematics
• Online resources such as Khan Academy and Mathway
• Calculators and online tools for solving equations
• Tutoring or online courses on algebra and mathematics

Q: Can I get help if I'm struggling with solving equations?

A: Yes, you can get help if you're struggling with solving equations. You can ask your teacher or tutor for assistance, or seek help online through resources such as Khan Academy and Mathway.

Conclusion

In this article, we answered some common questions related to solving the equation $1 = \frac{v+2}{v-4} + \frac{7v-42}{v-4}$. We discussed the main issue with the equation, the correct way to simplify it, and the importance of checking for inconsistencies. We also provided some resources for learning more about solving equations and offered tips for practicing and getting help.

Final Answer

The final answer is $\boxed{No solution}$.

Discussion

The equation $1 = \frac{v+2}{v-4} + \frac{7v-42}{v-4}$ is an example of an inconsistent equation, which means that there is no value of $v$ that satisfies the equation. This can occur when the equation is contradictory, such as when the left-hand side and right-hand side are equal to different values.

Related Topics

• Solving linear equations
• Simplifying fractions
• Algebraic techniques

References

• [1] "Algebra" by Michael Artin
• [2] "Linear Algebra" by Jim Hefferon
• [3] "Solving Equations" by Paul Dawkins