Solve For { X $} : : : { \frac{9}{x} = \frac{5}{x-3} \}

Mar 13, 2025 by ADMIN 56 views

**Solving for x in a Rational Equation**

Introduction

Rational equations are a type of algebraic equation that involves fractions with variables in the numerator or denominator. Solving for x in a rational equation can be a challenging task, but with the right approach, it can be done. In this article, we will explore how to solve for x in the rational equation $\frac{9}{x} = \frac{5}{x-3}$ .

Understanding the Equation

The given equation is $\frac{9}{x} = \frac{5}{x-3}$ . This equation involves two fractions with variables in the numerator and denominator. To solve for x, we need to eliminate the fractions and isolate the variable x.

Step 1: Cross-Multiply

The first step in solving this equation is to cross-multiply. This involves multiplying both sides of the equation by the denominators of the fractions. In this case, we will multiply both sides by x and x-3.

$\frac{9}{x} \cdot x = \frac{5}{x-3} \cdot (x-3)$

This simplifies to:

$9 = 5(x-3)$

Step 2: Distribute and Simplify

The next step is to distribute the 5 to the terms inside the parentheses and simplify the equation.

$9 = 5x - 15$

Step 3: Add 15 to Both Sides

To isolate the term with the variable, we need to add 15 to both sides of the equation.

$9 + 15 = 5x - 15 + 15$

This simplifies to:

$24 = 5x$

Step 4: Divide Both Sides by 5

Finally, we need to divide both sides of the equation by 5 to solve for x.

$\frac{24}{5} = \frac{5x}{5}$

This simplifies to:

$x = \frac{24}{5}$

Conclusion

Solving for x in a rational equation can be a challenging task, but with the right approach, it can be done. In this article, we explored how to solve for x in the rational equation $\frac{9}{x} = \frac{5}{x-3}$ . We used cross-multiplication, distribution, and simplification to isolate the variable x. The final solution is $x = \frac{24}{5}$ .

Example Use Cases

Rational equations are used in a variety of real-world applications, including:

Physics: Rational equations are used to describe the motion of objects under the influence of forces.
Engineering: Rational equations are used to design and optimize systems, such as electrical circuits and mechanical systems.
Economics: Rational equations are used to model economic systems and make predictions about future economic trends.

Tips and Tricks

When solving rational equations, it's essential to:

Check for extraneous solutions: Rational equations can have extraneous solutions, which are solutions that are not valid in the original equation.
Use a calculator: Rational equations can be complex and difficult to solve by hand. Using a calculator can help simplify the process.
Check the domain: Rational equations can have restrictions on the domain, which can affect the validity of the solution.

Common Mistakes

When solving rational equations, it's essential to avoid common mistakes, such as:

Not checking for extraneous solutions: Failing to check for extraneous solutions can lead to incorrect solutions.
Not using a calculator: Failing to use a calculator can make the process more difficult and increase the risk of errors.
Not checking the domain: Failing to check the domain can lead to invalid solutions.

Conclusion

Introduction

In our previous article, we explored how to solve for x in the rational equation $\frac{9}{x} = \frac{5}{x-3}$ . In this article, we will answer some common questions that readers may have about solving rational equations.

Q: What is a rational equation?

A: A rational equation is an equation that involves fractions with variables in the numerator or denominator.

Q: How do I know if an equation is a rational equation?

A: To determine if an equation is a rational equation, look for fractions with variables in the numerator or denominator. If you see a fraction with a variable in the numerator or denominator, it's likely a rational equation.

Q: What are some common types of rational equations?

A: Some common types of rational equations include:

Linear rational equations: These equations involve fractions with linear expressions in the numerator or denominator.
Quadratic rational equations: These equations involve fractions with quadratic expressions in the numerator or denominator.
Polynomial rational equations: These equations involve fractions with polynomial expressions in the numerator or denominator.

Q: How do I solve a rational equation?

A: To solve a rational equation, follow these steps:

Cross-multiply: Multiply both sides of the equation by the denominators of the fractions.
Distribute and simplify: Distribute the terms inside the parentheses and simplify the equation.
Add or subtract: Add or subtract the same value to both sides of the equation to isolate the variable.
Divide: Divide both sides of the equation by the coefficient of the variable to solve for x.

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

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

Not checking for extraneous solutions: Failing to check for extraneous solutions can lead to incorrect solutions.
Not using a calculator: Failing to use a calculator can make the process more difficult and increase the risk of errors.
Not checking the domain: Failing to check the domain can lead to invalid solutions.

Q: How do I check for extraneous solutions?

A: To check for extraneous solutions, plug the solution back into the original equation and check if it's true. If the solution is not true, it's an extraneous solution.

Q: What is the domain of a rational equation?

A: The domain of a rational equation is the set of all possible values of x that make the equation true. The domain can be restricted by the presence of fractions with variables in the numerator or denominator.

Q: How do I find the domain of a rational equation?

A: To find the domain of a rational equation, look for fractions with variables in the numerator or denominator. The domain is restricted by the values that make the denominator equal to zero.

Q: Can I use a calculator to solve rational equations?

A: Yes, you can use a calculator to solve rational equations. In fact, using a calculator can help simplify the process and reduce the risk of errors.

Q: Are there any online resources that can help me solve rational equations?

A: Yes, there are many online resources that can help you solve rational equations, including:

Math websites: Websites like Khan Academy, Mathway, and Wolfram Alpha offer step-by-step solutions to rational equations.
Online calculators: Online calculators like Desmos and Symbolab can help you solve rational equations and visualize the solutions.
Math software: Math software like Mathematica and Maple can help you solve rational equations and perform advanced mathematical operations.

Conclusion

Solving rational equations can be a challenging task, but with the right approach and resources, it can be done. In this article, we answered some common questions that readers may have about solving rational equations. We hope this article has been helpful in providing you with the information and resources you need to solve rational equations.