Solve For $a$.$\frac{3}{a} - \frac{4}{a+2} = 0$A) $ A = 1 2 A = \frac{1}{2} A = 2 1 [/tex] B) $a = 6$ C) $a = -\frac{6}{7}$ D) $ A = 2 A = 2 A = 2 [/tex]
Introduction
When solving rational equations, it's essential to understand the properties of fractions and how they interact with each other. In this problem, we're given the equation $\frac{3}{a} - \frac{4}{a+2} = 0$, and we need to solve for the variable $a$. This equation involves two fractions with different denominators, and we'll need to use algebraic techniques to simplify and solve it.
Step 1: Simplify the Equation
To simplify the equation, we can start by finding a common denominator for the two fractions. The common denominator is the product of the two denominators, which is $a(a+2)$. We can rewrite each fraction with this common denominator:
Now we can substitute these expressions back into the original equation:
Step 2: Eliminate the Common Denominator
Since the denominators are the same, we can eliminate them by multiplying both sides of the equation by $a(a+2)$. This will give us:
Step 3: Expand and Simplify
Next, we can expand the left-hand side of the equation by distributing the 3 to the terms inside the parentheses:
Step 4: Combine Like Terms
Now we can combine like terms by adding or subtracting the coefficients of the same variable:
Step 5: Solve for $a$
To solve for $a$, we can isolate the variable by subtracting 6 from both sides of the equation:
Step 6: Multiply by -1
Finally, we can multiply both sides of the equation by -1 to solve for $a$:
Conclusion
In this problem, we used algebraic techniques to simplify and solve a rational equation involving fractions. We found a common denominator, eliminated it, expanded and simplified the equation, combined like terms, and finally solved for the variable $a$. The solution to the equation is $a = 6$.
Comparison with Answer Choices
Now that we have solved the equation, we can compare our solution with the answer choices:
A) $a = \frac{1}{2}$
B) $a = 6$
C) $a = -\frac{6}{7}$
D) $a = 2$
Our solution, $a = 6$, matches answer choice B.
Final Answer
The final answer is:
Introduction
In our previous article, we solved the rational equation $\frac{3}{a} - \frac{4}{a+2} = 0$ and found that the solution is $a = 6$. However, we received several questions from readers regarding the solution and the steps involved in solving the equation. In this article, we'll address some of the most frequently asked questions and provide additional clarification on the solution.
Q: What is the common denominator of the two fractions?
A: The common denominator of the two fractions is the product of the two denominators, which is $a(a+2)$.
Q: Why did we multiply both sides of the equation by $a(a+2)$?
A: We multiplied both sides of the equation by $a(a+2)$ to eliminate the common denominator. This is a common technique used to simplify rational equations.
Q: Can we use other methods to solve the equation?
A: Yes, there are other methods to solve the equation, such as using the least common multiple (LCM) or factoring. However, the method we used in the previous article is a straightforward and efficient way to solve the equation.
Q: Why did we combine like terms in step 4?
A: We combined like terms in step 4 to simplify the equation and make it easier to solve. By combining like terms, we can eliminate unnecessary variables and make the equation more manageable.
Q: Can we solve the equation using a calculator?
A: Yes, we can solve the equation using a calculator. However, the calculator will only give us an approximate solution, whereas the algebraic method we used in the previous article gives us an exact solution.
Q: What if the equation has multiple solutions?
A: If the equation has multiple solutions, we can use the quadratic formula or other algebraic techniques to find all the solutions.
Q: Can we use the same method to solve other rational equations?
A: Yes, the method we used in the previous article can be applied to other rational equations involving fractions. However, the specific steps and techniques may vary depending on the equation.
Q: What if the equation has a denominator of zero?
A: If the equation has a denominator of zero, we need to be careful when simplifying the equation. We may need to use other techniques, such as factoring or the quadratic formula, to solve the equation.
Conclusion
In this article, we addressed some of the most frequently asked questions regarding the solution to the rational equation $\frac{3}{a} - \frac{4}{a+2} = 0$. We provided additional clarification on the solution and the steps involved in solving the equation. We also discussed other methods that can be used to solve the equation and addressed some common pitfalls that may arise when solving rational equations.
Additional Resources
For more information on solving rational equations, we recommend the following resources:
- Khan Academy: Rational Equations
- Mathway: Rational Equations
- Wolfram Alpha: Rational Equations
Final Answer
The final answer is: