Solve The Equation:${ \frac{x-14}{x-7} = -\frac{7}{x-7} - 8 }$Select The Correct Choice Below And, If Necessary, Fill In The Answer Box To Complete Your Choice.A. The Solution Set Is { {\ \square\ }$}$. (Simplify Your Answer.

Introduction

In this article, we will delve into solving a complex equation involving fractions. The equation is given as $\frac{x-14}{x-7} = -\frac{7}{x-7} - 8$. Our goal is to find the solution set for this equation, which will involve simplifying the expression and isolating the variable $x$. We will break down the solution into manageable steps, making it easier to understand and follow along.

Step 1: Simplify the Equation

To begin solving the equation, we need to simplify the expression on the right-hand side. We can start by combining the two fractions on the right-hand side.

$\frac{x-14}{x-7} = -\frac{7}{x-7} - 8$

We can rewrite the right-hand side as a single fraction by finding a common denominator.

$\frac{x-14}{x-7} = \frac{-7 - 8(x-7)}{x-7}$

Simplifying the numerator, we get:

$\frac{x-14}{x-7} = \frac{-7 - 8x + 56}{x-7}$

Combining like terms, we get:

$\frac{x-14}{x-7} = \frac{-8x + 49}{x-7}$

Now, we can see that the left-hand side and the right-hand side have a common denominator, which is $x-7$. We can eliminate the denominators by multiplying both sides of the equation by $x-7$.

$(x-14) = (-8x + 49)$

Step 2: Isolate the Variable

Now that we have eliminated the denominators, we can focus on isolating the variable $x$. We can start by adding $8x$ to both sides of the equation.

$x-14 + 8x = 49$

Combining like terms, we get:

$9x - 14 = 49$

Next, we can add $14$ to both sides of the equation.

$9x = 63$

Step 3: Solve for $x$

Now that we have isolated the variable $x$, we can solve for its value. We can divide both sides of the equation by $9$.

$x = \frac{63}{9}$

Simplifying the fraction, we get:

$x = 7$

Conclusion

In this article, we have solved the equation $\frac{x-14}{x-7} = -\frac{7}{x-7} - 8$. We broke down the solution into manageable steps, simplifying the expression and isolating the variable $x$. Our final answer is $x = 7$.

Discussion

The solution to this equation is $x = 7$. However, we need to check if this solution is valid by plugging it back into the original equation.

$\frac{7-14}{7-7} = -\frac{7}{7-7} - 8$

Simplifying the expression, we get:

$\frac{-7}{0} = -8$

This is an undefined expression, which means that the solution $x = 7$ is not valid.

Final Answer

The solution set for the equation $\frac{x-14}{x-7} = -\frac{7}{x-7} - 8$ is $\{\ \square\ \}$. This means that there is no valid solution for the equation.

References

• [1] "Algebra" by Michael Artin
• [2] "Calculus" by Michael Spivak

Note

Introduction

In our previous article, we solved the equation $\frac{x-14}{x-7} = -\frac{7}{x-7} - 8$. However, we found that the solution $x = 7$ was not valid. In this article, we will address some common questions and concerns related to solving this equation.

Q: What is the main concept behind solving this equation?

A: The main concept behind solving this equation is to simplify the expression and isolate the variable $x$. We used algebraic manipulations to combine like terms and eliminate the denominators.

Q: Why did we multiply both sides of the equation by $x-7$?

A: We multiplied both sides of the equation by $x-7$ to eliminate the denominators. This is a common technique used in algebra to simplify equations.

Q: What is the difference between a valid and invalid solution?

A: A valid solution is one that satisfies the original equation and does not result in an undefined expression. An invalid solution, on the other hand, is one that does not satisfy the original equation or results in an undefined expression.

Q: How can we check the validity of a solution?

A: We can check the validity of a solution by plugging it back into the original equation. If the expression is defined and satisfies the equation, then the solution is valid. If the expression is undefined or does not satisfy the equation, then the solution is invalid.

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

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

• Not checking the validity of solutions
• Not simplifying the expression before isolating the variable
• Not eliminating the denominators before solving for the variable
• Not plugging the solution back into the original equation to check its validity

Q: How can we apply the concepts learned from solving this equation to other problems?

A: The concepts learned from solving this equation can be applied to other problems involving algebraic manipulations, simplifying expressions, and isolating variables. These concepts are essential in solving a wide range of mathematical problems, including calculus and physics.

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

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

• Physics: Solving equations is essential in physics to describe the motion of objects, forces, and energies.
• Engineering: Solving equations is used in engineering to design and optimize systems, structures, and mechanisms.
• Economics: Solving equations is used in economics to model and analyze economic systems, markets, and behaviors.
• Computer Science: Solving equations is used in computer science to develop algorithms, models, and simulations.

Conclusion

In this article, we addressed some common questions and concerns related to solving the equation $\frac{x-14}{x-7} = -\frac{7}{x-7} - 8$. We emphasized the importance of checking the validity of solutions and avoiding common mistakes when solving equations. The concepts learned from solving this equation can be applied to other problems involvingic manipulations, simplifying expressions, and isolating variables.

References

• [1] "Algebra" by Michael Artin
• [2] "Calculus" by Michael Spivak
• [3] "Physics for Scientists and Engineers" by Paul A. Tipler
• [4] "Engineering Mathematics" by John Bird

Note

The concepts learned from solving this equation are essential in a wide range of mathematical and scientific fields. By understanding and applying these concepts, we can develop a deeper appreciation for the beauty and power of mathematics.