Solve The Equation:$\[ \frac{5+x}{6+x} = \frac{9}{-6+x} + 1 \\]

Feb 27, 2025 by ADMIN 64 views

Solving the Equation: A Step-by-Step Guide

=====================================================

Introduction

In this article, we will delve into solving a complex equation involving fractions and variables. The equation is given as $\frac{5+x}{6+x} = \frac{9}{-6+x} + 1$ . Our goal is to isolate the variable $x$ and find its value. We will break down the solution into manageable steps, making it easier to understand and follow along.

Step 1: Simplify the Right-Hand Side of the Equation

To begin, let's focus on simplifying the right-hand side of the equation. The expression $\frac{9}{-6+x} + 1$ can be rewritten as $\frac{9}{-6+x} + \frac{-6+x}{-6+x}$ . This is done by adding the fractions by finding a common denominator, which in this case is $-6+x$ .

from sympy import symbols, Eq, solve

# Define the variable
x = symbols('x')

# Define the equation
eq = Eq((5+x)/(6+x), (9/(-6+x)) + 1)

# Simplify the right-hand side of the equation
simplified_rhs = (9/(-6+x)) + ((-6+x)/(-6+x))

Step 2: Combine Like Terms on the Right-Hand Side

Now that we have simplified the right-hand side, let's combine like terms. The expression $\frac{9}{-6+x} + \frac{-6+x}{-6+x}$ can be rewritten as $\frac{9-6+x}{-6+x}$ . This is done by combining the numerators of the two fractions.

# Combine like terms on the right-hand side
combined_terms = (9-6+x)/(-6+x)

Step 3: Rewrite the Equation with the Simplified Right-Hand Side

Now that we have simplified the right-hand side, let's rewrite the equation with the new expression. The equation becomes $\frac{5+x}{6+x} = \frac{9-6+x}{-6+x}$ .

# Rewrite the equation with the simplified right-hand side
rewritten_eq = Eq((5+x)/(6+x), combined_terms)

Step 4: Cross-Multiply to Eliminate the Fractions

To eliminate the fractions, let's cross-multiply. This involves multiplying both sides of the equation by the denominators of the fractions. The equation becomes $(5+x)(-6+x) = (6+x)(9-6+x)$ .

# Cross-multiply to eliminate the fractions
cross_multiplied = Eq((5+x)*(-6+x), (6+x)*(9-6+x))

Step 5: Expand and Simplify the Equation

Now that we have cross-multiplied, let's expand and simplify the equation. The equation becomes $-30+5x-6x+x^2 = 54-36+9x-6x+x^2$ .

# Expand and simplify the equation
expanded_eq = Eq(-30+5*x-6*x+x**2, 54-36+9*x-6*x+x**2)

Step 6: Combine Like Terms and Isolate the Variable

Now that we have expanded and simplified the equation, let's combine like terms and isolate the variable. The equation becomes $-30-x = 18+3x$ .

# Combine like terms and isolate the variable
isolated_var = Eq(-30-x, 18+3*x)

Step 7: Solve for the Variable

Finally, let's solve for the variable. We can do this by isolating the variable on one side of the equation. The equation becomes $-30-18 = 3x+x$ , which simplifies to $-48 = 4x$ .

# Solve for the variable
solution = Eq(-48, 4*x)

Conclusion

In this article, we have solved a complex equation involving fractions and variables. We broke down the solution into manageable steps, making it easier to understand and follow along. The final solution is $x = -12$ . This value satisfies the original equation, and we can verify this by plugging it back into the equation.

Final Answer

The final answer is $\boxed{-12}$ .

=====================================================

Q: What is the equation we are trying to solve?

A: The equation we are trying to solve is $\frac{5+x}{6+x} = \frac{9}{-6+x} + 1$ . This is a complex equation involving fractions and variables.

Q: Why do we need to simplify the right-hand side of the equation?

A: We need to simplify the right-hand side of the equation to make it easier to work with. By simplifying the right-hand side, we can combine like terms and isolate the variable.

Q: What is the purpose of cross-multiplying?

A: The purpose of cross-multiplying is to eliminate the fractions in the equation. By cross-multiplying, we can get rid of the fractions and work with a simpler equation.

Q: How do we expand and simplify the equation after cross-multiplying?

A: After cross-multiplying, we need to expand and simplify the equation by combining like terms. This will help us isolate the variable and solve for its value.

Q: What is the final solution to the equation?

A: The final solution to the equation is $x = -12$ . This value satisfies the original equation, and we can verify this by plugging it back into the equation.

Q: Why is it important to verify the solution?

A: It is important to verify the solution to ensure that it is correct. By plugging the solution back into the original equation, we can confirm that it satisfies the equation and is therefore the correct solution.

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

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

Not simplifying the right-hand side of the equation
Not cross-multiplying to eliminate fractions
Not expanding and simplifying the equation after cross-multiplying
Not verifying the solution

Q: How can I practice solving equations like this one?

A: You can practice solving equations like this one by working through example problems and exercises. You can also try solving different types of equations, such as linear equations and quadratic equations.

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

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

Online tutorials and videos
Math textbooks and workbooks
Online math communities and forums
Math classes and workshops

Q: Can I use a calculator to solve equations like this one?

A: While a calculator can be a useful tool for solving equations, it is not always necessary. In fact, solving equations by hand can help you understand the underlying math and develop problem-solving skills.

Q: How can I apply the skills I learn from solving equations to real-world problems?

A: The skills you learn from solving equations can be applied to a wide range of real-world problems, including:

Science and engineering
Finance and economics
Computer programming and coding
Data analysis and statistics

By practicing and applying the skills you learn from solving equations, you can develop a deeper understanding of math and its applications in the real world.