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

Introduction

Solving equations involving fractions can be a challenging task, especially when dealing with variables in the denominator. In this problem, we are given an equation with fractions and a variable in the denominator, and we need to solve for the value of the variable. The equation is $\frac{1}{3a^2} - \frac{1}{a} = \frac{1}{6a^2}$, and we need to find the value of $a$ that satisfies this equation.

Step 1: Multiply both sides of the equation by the least common multiple (LCM) of the denominators

To eliminate the fractions, we need to multiply both sides of the equation by the least common multiple (LCM) of the denominators. The LCM of $3a^2$, $a$, and $6a^2$ is $6a^2$. Multiplying both sides of the equation 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: Simplify the equation

Simplifying the equation, we get:

$$2 - 6a = 1$$

Step 3: Isolate the variable

To isolate the variable $a$, we need to get all the terms with $a$ on one side of the equation. Subtracting 1 from both sides of the equation gives us:

$$2 - 1 - 6a = 1 - 1$$

$$-6a = 0$$

Step 4: Solve for the variable

To solve for the variable $a$, we need to get rid of the coefficient $-6$. Dividing both sides of the equation by $-6$ gives us:

$$\frac{-6a}{-6} = \frac{0}{-6}$$

$$a = 0$$

Conclusion

In this problem, we were given an equation with fractions and a variable in the denominator, and we needed to solve for the value of the variable. By multiplying both sides of the equation by the least common multiple of the denominators, simplifying the equation, isolating the variable, and solving for the variable, we found that the value of $a$ that satisfies the equation is $a = 0$.

Final Answer

The final answer is $\boxed{0}$.

Discussion

This problem is a good example of how to solve equations involving fractions and variables in the denominator. By following the steps outlined above, we can eliminate the fractions and isolate the variable to find the solution. However, it's worth noting that the solution $a = 0$ may not be valid in all cases, as it would result in division by zero in the original equation. Therefore, it's always a good idea to check the solution and make sure it's valid before accepting it as the final answer.

Related Problems

If you're interested in practicing more problems like this, here are a few related problems:

• Solve the equation $\frac{1}{2a} - \frac{1}{3a} = \frac{1}{6a}$
• Solve the equation $\frac{1}{a^2} - \frac{1}{2a} = \frac{1}{3a^2}$
• Solve the equation $\frac{1}{3a} - \frac{1}{2a} = \frac{1}{6a}$

These problems involve similar techniques and concepts, and can help you build your skills and confidence in solving equations involving fractions and variables in the denominator.

Introduction

Solving equations involving fractions and variables in the denominator can be a challenging task, but with the right techniques and strategies, it can be done. In this article, we will answer some common questions related to solving equations involving fractions and variables in the denominator.

Q: What is the first step in solving an equation involving fractions and variables in the denominator?

A: The first step in solving an equation involving fractions and variables in the denominator is to multiply both sides of the equation by the least common multiple (LCM) of the denominators. This will eliminate the fractions and make it easier to solve the equation.

Q: How do I find the LCM of the denominators?

A: To find the LCM of the denominators, you need to list the multiples of each denominator and find the smallest multiple that is common to all of them. For example, if the denominators are 2, 3, and 4, the LCM would be 12.

Q: What if the LCM is not a whole number?

A: If the LCM is not a whole number, you can multiply both sides of the equation by the smallest power of 10 that will make the LCM a whole number. For example, if the LCM is 2.5, you can multiply both sides of the equation by 2 to make the LCM a whole number.

Q: How do I simplify the equation after multiplying both sides by the LCM?

A: After multiplying both sides of the equation by the LCM, you can simplify the equation by combining like terms and canceling out any common factors.

Q: What if the equation has multiple variables?

A: If the equation has multiple variables, you can use the same techniques and strategies as before to solve for one variable at a time. For example, if the equation is 2x + 3y = 5, you can solve for x first and then substitute the value of x into the equation to solve for y.

Q: How do I check my solution?

A: To check your solution, you can plug the value of the variable back into the original equation and make sure that it is true. For example, if the solution is x = 2, you can plug x = 2 into the equation 2x + 3y = 5 and make sure that it is true.

Q: What if my solution is not valid?

A: If your solution is not valid, it may be because you made a mistake in solving the equation or because the equation has no solution. In this case, you can go back and recheck your work to make sure that you made no mistakes. If you are still having trouble, you can try using a different method or seeking help from a teacher or tutor.

Q: Are there any other tips or strategies for solving equations involving fractions and variables in the denominator?

A: Yes, here are a few additional tips and strategies that may be helpful:

• Make sure to read the problem carefully and understand what is being asked.
• Use a pencil and paper to work out the problem step by step.
• Check your work as you go to make sure that you are making no mistakes.
• Don't be afraid to ask for help if you are having trouble.
• Practice, practice, practice! The more you practice solving equations involving fractions and variables in the denominator, the more comfortable you will become with the techniques and strategies.

Conclusion

Solving equations involving fractions and variables in the denominator can be a challenging task, but with the right techniques and strategies, it can be done. By following the steps outlined in this article and practicing regularly, you can become more confident and proficient in solving these types of equations. Remember to always read the problem carefully, use a pencil and paper to work out the problem step by step, and check your work as you go to make sure that you are making no mistakes.