Understanding the Concept of Clearing Fractions

Oct 13, 2025 by miodeya.com 17 views

Introduction

When dealing with rational equations, one of the most common challenges is clearing fractions. This process involves eliminating the denominators of the fractions to simplify the equation and make it easier to solve. In this article, we will focus on solving rational equations by clearing fractions, using the given equation $\frac{9}{u+7} = \frac{\square}{(u-5)(u+7)}$ as an example.

Understanding the Concept of Clearing Fractions

Clearing fractions is a crucial step in solving rational equations. It involves multiplying both sides of the equation by the least common multiple (LCM) of the denominators to eliminate the fractions. The LCM is the smallest multiple that all the denominators have in common.

Identifying the Least Common Multiple (LCM)

To clear fractions, we need to identify the LCM of the denominators. In the given equation, the denominators are $(u+7)$ and $(u-5)(u+7)$ . The LCM of these denominators is $(u-5)(u+7)$ , which is the product of the two denominators.

Multiplying Both Sides by the LCM

Now that we have identified the LCM, we can multiply both sides of the equation by $(u-5)(u+7)$ to clear the fractions.

$\frac{9}{u+7} = \frac{\square}{(u-5)(u+7)}$

Multiplying both sides by $(u-5)(u+7)$ :

$9(u-5)(u+7) = \square(u+7)$

Expanding and Simplifying the Equation

Now that we have cleared the fractions, we can expand and simplify the equation.

$9(u-5)(u+7) = \square(u+7)$

Expanding the left-hand side:

$9(u^2 + 2u - 35) = \square(u+7)$

Simplifying the left-hand side:

$9u^2 + 18u - 315 = \square(u+7)$

Equating the Numerators

Now that we have simplified the equation, we can equate the numerators.

$9u^2 + 18u - 315 = \square(u+7)$

Since the denominators are the same, we can equate the numerators:

$9u^2 + 18u - 315 = \square$

Solving for the Unknown

Now that we have equated the numerators, we can solve for the unknown.

$9u^2 + 18u - 315 = \square$

Subtracting $\square$ from both sides:

$9u^2 + 18u - 315 - \square = 0$

Factoring the Quadratic Equation

Now that we have a quadratic equation, we can factor it.

$9u^2 + 18u - 315 - \square = 0$

Factoring the quadratic equation:

$(3u - 21)(3u + 15) = \square$

Solving for the Unknown

Now that we have factored the quadratic equation, we can solve for the unknown.

$(3u - 21)(3u + 15) = \square$

Expanding the left-hand side:

$9u^2 + 45u - 63u - 315 = \square$

Simplifying the left-hand side:

$9u^2 - 18u - 315 = \square$

Equating the Numerators

Now that we have simplified the equation, we can equate the numerators.

$9u^2 - 18u - 315 = \square$

Since the denominators are the same, we can equate the numerators:

$9u^2 - 18u - 315 = 0$

Solving for the Unknown

Now that we have equated the numerators, we can solve for the unknown.

$9u^2 - 18u - 315 = 0$

Dividing both sides by 9:

$u^2 - 2u - 35 = 0$

Factoring the Quadratic Equation

Now that we have a quadratic equation, we can factor it.

$u^2 - 2u - 35 = 0$

Factoring the quadratic equation:

$(u - 7)(u + 5) = 0$

Solving for the Unknown

Now that we have factored the quadratic equation, we can solve for the unknown.

$(u - 7)(u + 5) = 0$

Using the zero-product property:

$u - 7 = 0$ or $u + 5 = 0$

Solving for the Unknown

Now that we have two possible solutions, we can solve for the unknown.

$u - 7 = 0$

Adding 7 to both sides:

$u = 7$

$u + 5 = 0$

Subtracting 5 from both sides:

$u = -5$

Conclusion

In this article, we have learned how to solve rational equations by clearing fractions. We have used the given equation $\frac{9}{u+7} = \frac{\square}{(u-5)(u+7)}$ as an example and have walked through the steps of clearing fractions, expanding and simplifying the equation, equating the numerators, and solving for the unknown. We have also learned how to factor quadratic equations and use the zero-product property to solve for the unknown. With these skills, you can now solve rational equations with ease and confidence.

Frequently Asked Questions

Q: What is the least common multiple (LCM) of the denominators? A: The LCM of the denominators is the product of the two denominators.
Q: How do I clear fractions in a rational equation? A: To clear fractions, multiply both sides of the equation by the LCM of the denominators.
Q: What is the zero-product property? A: The zero-product property states that if the product of two or more factors is equal to zero, then at least one of the factors must be equal to zero.

Final Thoughts

Solving rational equations can be challenging, but with the right skills and techniques, it can be a breeze. By clearing fractions, expanding and simplifying the equation, equating the numerators, and solving for the unknown, you can solve rational equations with ease and confidence. Remember to always identify the LCM of the denominators, multiply both sides of the equation by the LCM, and use the zero-product property to solve for the unknown. With practice and patience, you will become a master of solving rational equations.

Introduction

Solving rational equations can be a challenging task, but with the right skills and techniques, it can be a breeze. In this article, we will provide a Q&A guide to help you solve rational equations with ease and confidence.

Q: What is a rational equation?

A: A rational equation is an equation that contains one or more rational expressions, which are expressions that contain fractions.

Q: What is the least common multiple (LCM) of the denominators?

A: The LCM of the denominators is the product of the two denominators.

Q: How do I clear fractions in a rational equation?

A: To clear fractions, multiply both sides of the equation by the LCM of the denominators.

Q: What is the zero-product property?

A: The zero-product property states that if the product of two or more factors is equal to zero, then at least one of the factors must be equal to zero.

Q: How do I solve a rational equation?

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

Identify the LCM of the denominators.
Multiply both sides of the equation by the LCM.
Expand and simplify the equation.
Equate the numerators.
Solve for the unknown.

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

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

Not identifying the LCM of the denominators.
Not multiplying both sides of the equation by the LCM.
Not expanding and simplifying the equation.
Not equating the numerators.
Not solving for the unknown.

Q: How do I factor a quadratic equation?

A: To factor a quadratic equation, follow these steps:

Look for two numbers whose product is equal to the constant term and whose sum is equal to the coefficient of the linear term.
Write the quadratic equation as a product of two binomials.
Simplify the expression.

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

A: A rational expression is an expression that contains fractions, while a rational equation is an equation that contains one or more rational expressions.

Q: How do I simplify a rational expression?

A: To simplify a rational expression, follow these steps:

Factor the numerator and denominator.
Cancel out any common factors.
Simplify the expression.

Q: What are some common applications of rational equations?

A: Some common applications of rational equations include:

Physics: Rational equations are used to describe the motion of objects.
Engineering: Rational equations are used to design and analyze systems.
Economics: Rational equations are used to model economic systems.

Q: How do I use rational equations in real-world problems?

A: To use rational equations in real-world problems, follow these steps:

Identify the problem and the variables involved.
Write an equation that models the problem.
Solve the equation using rational equations.
Interpret the results in the context of the problem.

Q: What are some common mistakes to avoid when using rational equations in real-world problems?

A: Some common mistakes to avoid when using rational equations in real-world problems include:

Not identifying the problem and the variables involved.
Not writing an equation that models the problem.
Not solving the equation using rational equations.
Not interpreting the results in the context of the problem.

Conclusion

Solving rational equations can be a challenging task, but with the right skills and techniques, it can be a breeze. By following the steps outlined in this Q&A guide, you can solve rational equations with ease and confidence. Remember to always identify the LCM of the denominators, multiply both sides of the equation by the LCM, and use the zero-product property to solve for the unknown. With practice and patience, you will become a master of solving rational equations.

Final Thoughts

Rational equations are a powerful tool for solving problems in a variety of fields, including physics, engineering, and economics. By mastering the skills and techniques outlined in this Q&A guide, you can use rational equations to solve real-world problems with ease and confidence. Remember to always identify the LCM of the denominators, multiply both sides of the equation by the LCM, and use the zero-product property to solve for the unknown. With practice and patience, you will become a master of solving rational equations.