Solve: $\frac{1}{3a^2} - \frac{1}{a} = \frac{1}{6a^2}$The Solution Is $a =$

Introduction

Rational equations are a fundamental concept in algebra, and solving them requires a deep understanding of fractions, exponents, and algebraic manipulations. In this article, we will focus on solving a specific rational equation, $\frac{1}{3a^2} - \frac{1}{a} = \frac{1}{6a^2}$, and derive the solution for the variable $a$.

Understanding Rational Equations

A rational equation is an equation that contains one or more rational expressions, which are fractions of polynomials. Rational expressions can be added, subtracted, multiplied, and divided, just like regular fractions. However, when solving rational equations, we need to be careful about the signs and the order of operations.

Step 1: Multiply Both Sides by the Least Common Multiple (LCM)

To solve the given rational equation, we need to eliminate the fractions by multiplying both sides by the least common multiple (LCM) of the denominators. In this case, the LCM is $6a^2$. Multiplying both sides by $6a^2$ gives us:

$$6a^2 \left( \frac{1}{3a^2} - \frac{1}{a} \right) = 6a^2 \left( \frac{1}{6a^2} \right)$$

Step 2: Distribute and Simplify

Now, we can distribute the $6a^2$ to the terms inside the parentheses:

$$2 - 2a = 1$$

Step 3: Isolate the Variable

Next, we need to isolate the variable $a$ by moving all the terms to one side of the equation:

$$2 - 1 = 2a$$

$$1 = 2a$$

Step 4: Solve for $a$

Finally, we can solve for $a$ by dividing both sides by 2:

$$a = \frac{1}{2}$$

Conclusion

In this article, we solved the rational equation $\frac{1}{3a^2} - \frac{1}{a} = \frac{1}{6a^2}$ by multiplying both sides by the LCM, distributing and simplifying, isolating the variable, and solving for $a$. The solution to the equation is $a = \frac{1}{2}$.

Tips and Tricks

When solving rational equations, it's essential to:

• Identify the LCM of the denominators and multiply both sides by it.
• Distribute and simplify the equation carefully.
• Isolate the variable by moving all the terms to one side of the equation.
• Solve for the variable by performing the necessary operations.

By following these steps and tips, you can confidently solve rational equations and derive the solution for the variable.

Real-World Applications

Rational equations have numerous real-world applications in various fields, such as:

• Physics: Rational equations are used to describe the motion of objects, including velocity, acceleration, and force.
• Engineering: Rational equations are used to design and optimize systems, including electrical circuits, mechanical systems, and control systems.
• Economics: Rational equations are used to model economic systems, including supply and demand, inflation, and interest rates.

Introduction

In our previous article, we solved the rational equation $\frac{1}{3a^2} - \frac{1}{a} = \frac{1}{6a^2}$ and derived the solution for the variable $a$. However, we know that practice makes perfect, and solving rational equations can be a challenging task. In this article, we will provide a Q&A guide to help you better understand the concepts and techniques involved in solving rational equations.

Q: What is a rational equation?

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

Q: How do I identify the least common multiple (LCM) of the denominators?

A: To identify the LCM, you need to find the smallest multiple that all the denominators have in common. For example, if the denominators are $3a^2$, $a$, and $6a^2$, the LCM is $6a^2$.

Q: Why do I need to multiply both sides by the LCM?

A: Multiplying both sides by the LCM eliminates the fractions and allows you to simplify the equation.

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

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

Q: How do I distribute and simplify the equation?

A: To distribute and simplify the equation, you need to multiply the terms inside the parentheses by the factor outside the parentheses.

Q: What is the order of operations when solving rational equations?

A: The order of operations is:

1. Multiply both sides by the LCM.
2. Distribute and simplify the equation.
3. Isolate the variable.
4. Solve for the variable.

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

A: Some common mistakes to avoid include:

• Not identifying the LCM correctly.
• Not distributing and simplifying the equation correctly.
• Not isolating the variable correctly.
• Not solving for the variable correctly.

Q: How do I check my solution?

A: To check your solution, you need to plug the value of the variable back into the original equation and verify that it is true.

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

A: Rational equations have numerous real-world applications in various fields, including:

• Physics: Rational equations are used to describe the motion of objects, including velocity, acceleration, and force.
• Engineering: Rational equations are used to design and optimize systems, including electrical circuits, mechanical systems, and control systems.
• Economics: Rational equations are used to model economic systems, including supply and demand, inflation, and interest rates.

Conclusion

Solving rational equations can be a challenging task, but with practice and patience, you can become proficient in solving these types of equations. By following the steps outlined in this article and avoiding common mistakes, you can confidently solve rational equations and derive the solution for the variable.

Additional Resources

For further practice and review, we recommend the following resources:

• Khan Academy: Rational Equations
• Mathway: Rational Equations
• Wolfram Alpha: Rational Equations

By using these resources and practicing regularly, you can improve your skills and become proficient in solving rational equations.