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

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Introduction

Rational equations are a fundamental concept in algebra, and solving them requires a deep understanding of fractions, exponents, and algebraic manipulation. In this article, we will focus on solving a specific rational equation, 13a2−1a=16a2\frac{1}{3a^2} - \frac{1}{a} = \frac{1}{6a^2}, and provide a step-by-step guide on how to arrive at the solution.

Understanding Rational Equations

A rational equation is an equation that contains one or more rational expressions, which are fractions that contain variables in the numerator or denominator. Rational equations can be solved using various techniques, including factoring, cross-multiplication, and algebraic manipulation.

The Given Equation

The given equation is 13a2−1a=16a2\frac{1}{3a^2} - \frac{1}{a} = \frac{1}{6a^2}. To solve this equation, we need to first simplify it by finding a common denominator.

Step 1: Simplify the Equation

To simplify the equation, we need to find a common denominator for the three fractions. The least common multiple (LCM) of 3a23a^2, aa, and 6a26a^2 is 6a26a^2. We can rewrite each fraction with the common denominator:

13a2=26a2\frac{1}{3a^2} = \frac{2}{6a^2}

1a=66a2\frac{1}{a} = \frac{6}{6a^2}

16a2=16a2\frac{1}{6a^2} = \frac{1}{6a^2}

Now, we can rewrite the equation as:

26a2−66a2=16a2\frac{2}{6a^2} - \frac{6}{6a^2} = \frac{1}{6a^2}

Step 2: Combine Like Terms

We can combine the two fractions on the left-hand side of the equation by subtracting their numerators:

2−66a2=16a2\frac{2-6}{6a^2} = \frac{1}{6a^2}

This simplifies to:

−46a2=16a2\frac{-4}{6a^2} = \frac{1}{6a^2}

Step 3: Eliminate the Common Denominator

Since both fractions have the same denominator, we can eliminate the denominator by multiplying both sides of the equation by 6a26a^2:

−4=1-4 = 1

This is a contradiction, as −4-4 is not equal to 11. Therefore, the original equation has no solution.

Conclusion

In this article, we solved the rational equation 13a2−1a=16a2\frac{1}{3a^2} - \frac{1}{a} = \frac{1}{6a^2} using a step-by-step guide. We simplified the equation by finding a common denominator, combined like terms, and eliminated the common denominator. The final result was a contradiction, indicating that the original equation has no solution.

Common Mistakes to Avoid

When solving rational equations, there are several common mistakes to avoid:

  • Not finding a common denominator: Failing to find a common denominator can lead to incorrect solutions.
  • Not combining like terms: Failing to combine like terms can lead to incorrect solutions.
  • Not eliminating the common denominator: Failing to eliminate the common denominator can lead to incorrect solutions.

Tips and Tricks

When solving rational equations, here are some tips and tricks to keep in mind:

  • Use a common denominator: Finding a common denominator can simplify the equation and make it easier to solve.
  • Combine like terms: Combining like terms can simplify the equation and make it easier to solve.
  • Eliminate the common denominator: Eliminating the common denominator can simplify the equation and make it easier to solve.

Real-World Applications

Rational equations have numerous real-world applications, including:

  • Physics: Rational equations are used to describe the motion of objects and the behavior of physical systems.
  • 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 economic trends.

Conclusion

Introduction

In our previous article, we solved the rational equation 13a2−1a=16a2\frac{1}{3a^2} - \frac{1}{a} = \frac{1}{6a^2} using a step-by-step guide. 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 that contain variables in the numerator or denominator.

Q: How do I simplify a rational equation?

A: To simplify a rational equation, you need to find a common denominator for the fractions. The least common multiple (LCM) of the denominators is the common denominator. Once you have found the common denominator, you can rewrite each fraction with the common denominator and combine like terms.

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

A: The least common multiple (LCM) of a set of numbers is the smallest number that is a multiple of each of the numbers in the set. In the case of rational equations, the LCM is the common denominator.

Q: How do I combine like terms in a rational equation?

A: To combine like terms in a rational equation, you need to add or subtract the numerators of the fractions that have the same denominator. For example, if you have two fractions with the same denominator, you can combine them by adding or subtracting their numerators.

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

A: A rational expression is a fraction that contains variables in the numerator or denominator. A rational equation is an equation that contains one or more rational expressions.

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 hand to make sure that the calculator is giving you the correct answer.

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

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

  • Not finding a common denominator
  • Not combining like terms
  • Not eliminating the common denominator
  • Not checking your work by hand

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

A: To determine if a rational equation has a solution, you need to check if the equation is true for all values of the variable. If the equation is true for all values of the variable, then the equation has a solution. If the equation is not true for all values of the variable, then the equation does not have a solution.

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

A: Yes, you can use rational equations to model real-world problems. Rational equations are used in a variety of fields, including physics, engineering, and economics.

Q: What are some examples of rational equations in real-world problems?

A: Some examples of rational equations in real-world problems include:

  • Modeling the motion of objects in physics
  • Designing and optimizing systems in engineering
  • Modeling economic systems and making predictions about economic trends

Conclusion

In conclusion, solving rational equations requires a deep understanding of fractions, exponents, and algebraic manipulation. By following a step-by-step guide and avoiding common mistakes, you can arrive at the solution to a rational equation. Remember to use a common denominator, combine like terms, and eliminate the common denominator to simplify the equation and make it easier to solve.