Steps to Solve Rational Equations

Oct 13, 2025 by talynt.org 98 views

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Introduction

Rational equations are a fundamental concept in algebra, and solving them requires a clear understanding of the underlying principles. In this article, we will delve into the world of rational equations and provide a step-by-step guide on how to solve them. We will also apply this knowledge to solve a specific rational equation and choose the correct answer from the given options.

What are Rational Equations?

Rational equations are equations that contain rational expressions, which are fractions of polynomials. Rational expressions are formed by dividing two polynomials, and they can be simplified or factored to solve the equation. Rational equations can be linear or non-linear, and they can have one or more variables.

Steps to Solve Rational Equations

Solving rational equations involves several steps:

Cross-multiply: The first step in solving a rational equation is to cross-multiply, which means multiplying the numerator of the first fraction by the denominator of the second fraction, and vice versa.
Simplify: After cross-multiplying, simplify the resulting equation by combining like terms.
Factor: Factor the numerator and denominator of the resulting equation, if possible.
Solve: Solve the resulting equation for the variable.
Check: Check the solution by plugging it back into the original equation.

Solving the Rational Equation

Now, let's apply these steps to solve the rational equation:

{ \frac{x+6}{x-3} = \frac{4}{7} \}

Step 1: Cross-multiply

Cross-multiplying the equation, we get:

{ 7(x+6) = 4(x-3) \}

Step 2: Simplify

Simplifying the equation, we get:

{ 7x + 42 = 4x - 12 \}

Step 3: Combine like terms

Combining like terms, we get:

{ 3x = -54 \}

Step 4: Solve

Solving for x, we get:

{ x = -18 \}

Step 5: Check

Checking the solution by plugging it back into the original equation, we get:

{ \frac{-18+6}{-18-3} = \frac{4}{7} \}

{ \frac{-12}{-21} = \frac{4}{7} \}

{ \frac{4}{7} = \frac{4}{7} \}

The solution checks out!

Conclusion

In conclusion, solving rational equations requires a clear understanding of the underlying principles and a step-by-step approach. By cross-multiplying, simplifying, factoring, solving, and checking, we can solve rational equations and choose the correct answer from the given options.

Final Answer

The final answer is:

D. x = -18

This is the correct answer to the rational equation:

{ \frac{x+6}{x-3} = \frac{4}{7} \}

I hope this article has provided a clear understanding of how to solve rational equations and choose the correct answer from the given options.

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Introduction

In our previous article, we discussed how to solve rational equations and provided a step-by-step guide on how to approach these types of problems. However, we know that there are many more questions that you may have about rational equations. In this article, we will address some of the most frequently asked questions about rational equations and provide clear and concise answers.

Q&A

Q: What is a rational equation?

A: A rational equation is an equation that contains rational expressions, which are fractions of polynomials.

Q: What is the difference between a rational equation and a rational expression?

A: A rational expression is a fraction of polynomials, while a rational equation is an equation that contains rational expressions.

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

A: An equation is a rational equation if it contains fractions of polynomials. Look for fractions with polynomials in the numerator and denominator.

Q: What are some common types of rational equations?

A: Some common types of rational equations include:

Linear rational equations: These are rational equations where the numerator and denominator are linear expressions.
Non-linear rational equations: These are rational equations where the numerator and denominator are non-linear expressions.
Equations with multiple variables: These are rational equations that contain multiple variables.

Q: How do I solve a rational equation with multiple variables?

A: To solve a rational equation with multiple variables, you can use the same steps as solving a rational equation with one variable. However, you may need to use substitution or elimination methods to solve for the variables.

Q: What is the most important thing to remember when solving rational equations?

A: The most important thing to remember when solving rational equations is to cross-multiply and simplify the equation before solving for the variable.

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

A: Yes, you can use a calculator to solve rational equations. However, it's always a good idea to check your work by plugging the solution back into the original equation.

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

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

Not cross-multiplying and simplifying the equation before solving for the variable.
Not checking the solution by plugging it back into the original equation.
Not considering the domain of the rational expression.

Q: How do I know if a rational equation has a solution?

A: A rational equation has a solution if the equation is true for at least one value of the variable. You can check if the equation has a solution by plugging in different values of the variable and seeing if the equation is true.

Q: Can I use rational equations to solve real-world problems?

A: Yes, rational equations can be used to solve real-world problems. For example, you can use rational equations to model population growth, chemical reactions, and electrical circuits.

Conclusion

In conclusion, rational equations are an important concept in algebra, and solving them requires a clear understanding of the underlying principles. By following the steps outlined in this article and avoiding common mistakes, you can solve rational equations and apply them to real-world problems.

Final Tips

Always cross-multiply and simplify the equation before solving for the variable.
Check your work by plugging the solution back into the original equation.
Consider the domain of the rational expression when solving rational equations.
Use rational equations to solve real-world problems and model complex phenomena.

We hope this article has provided a clear understanding of rational equations and how to solve them. If you have any further questions or need additional help, please don't hesitate to ask.