Solve For $y$.$\frac{5}{2y+12} - 3 = -\frac{2}{y+6}$If There Is More Than One Solution, Separate Them With Commas. If There Is No Solution, State No Solution.
Introduction
When it comes to solving rational equations, one of the most common challenges is isolating the variable. In this article, we will focus on solving the equation $\frac{5}{2y+12} - 3 = -\frac{2}{y+6}$ for the variable $y$. We will break down the solution into manageable steps, using algebraic techniques to simplify the equation and isolate the variable.
Step 1: Simplify the Equation
The first step in solving the equation is to simplify it by getting rid of the fractions. To do this, we can multiply both sides of the equation by the least common multiple (LCM) of the denominators, which is $(2y+12)(y+6)$. This will eliminate the fractions and make it easier to work with the equation.
\frac{5}{2y+12} - 3 = -\frac{2}{y+6}
Step 2: Multiply Both Sides by the LCM
Now that we have identified the LCM, we can multiply both sides of the equation by $(2y+12)(y+6)$. This will eliminate the fractions and give us a simpler equation to work with.
(2y+12)(y+6)\left(\frac{5}{2y+12} - 3\right) = (2y+12)(y+6)\left(-\frac{2}{y+6}\right)
Step 3: Distribute and Simplify
After multiplying both sides of the equation by the LCM, we can distribute and simplify the equation. This will give us a simpler equation to work with and make it easier to isolate the variable.
5(y+6) - 3(2y+12) = -2(2y+12)
Step 4: Expand and Simplify
Now that we have distributed and simplified the equation, we can expand and simplify it further. This will give us a simpler equation to work with and make it easier to isolate the variable.
5y + 30 - 6y - 36 = -4y - 24
Step 5: Combine Like Terms
After expanding and simplifying the equation, we can combine like terms. This will give us a simpler equation to work with and make it easier to isolate the variable.
-y - 6 = -4y - 24
Step 6: Add 4y to Both Sides
Now that we have combined like terms, we can add 4y to both sides of the equation. This will give us a simpler equation to work with and make it easier to isolate the variable.
5y = -18
Step 7: Divide Both Sides by 5
Finally, we can divide both sides of the equation by 5 to solve for y. This will give us the final solution to the equation.
y = -\frac{18}{5}
Conclusion
In this article, we have solved the equation $\frac{5}{2y+12} - 3 = -\frac{2}{y+6}$ for the variable $y$. We have broken down the solution into manageable steps, using algebraic techniques to simplify the equation and isolate the variable. The final solution to the equation is $y = -\frac{18}{5}$.
Discussion
The solution to the equation $\frac{5}{2y+12} - 3 = -\frac{2}{y+6}$ is $y = -\frac{18}{5}$. This solution is valid for all values of y that satisfy the original equation. However, it is worth noting that the original equation may have multiple solutions, depending on the values of the variables involved. In this case, we have found a single solution to the equation, which is $y = -\frac{18}{5}$.
Final Answer
The final answer to the equation $\frac{5}{2y+12} - 3 = -\frac{2}{y+6}$ is $y = -\frac{18}{5}$.
Introduction
Solving rational equations can be a challenging task, especially when dealing with complex fractions and multiple variables. In this article, we will address some of the most frequently asked questions (FAQs) about solving rational equations, providing step-by-step explanations and examples to help you better understand the concepts.
Q: What is a rational equation?
A: A rational equation is an equation that contains one or more fractions, where the numerator and denominator are polynomials. Rational equations can be linear or nonlinear, and they often involve variables in the numerator and denominator.
Q: How do I simplify a rational equation?
A: To simplify a rational equation, you can start by factoring the numerator and denominator, if possible. Then, you can cancel out any common factors between the numerator and denominator. Finally, you can multiply both sides of the equation by the least common multiple (LCM) of the denominators to eliminate the fractions.
Q: What is the least common multiple (LCM) of the denominators?
A: The LCM of the denominators is the smallest multiple that all the denominators have in common. To find the LCM, you can list the multiples of each denominator and find the smallest multiple that appears in all the lists.
Q: How do I multiply both sides of the equation by the LCM?
A: To multiply both sides of the equation by the LCM, you can simply multiply each term on both sides of the equation by the LCM. This will eliminate the fractions and make it easier to work with the equation.
Q: What is the difference between a rational equation and a rational expression?
A: A rational equation is an equation that contains one or more fractions, where the numerator and denominator are polynomials. A rational expression, on the other hand, is an expression that contains one or more fractions, where the numerator and denominator are polynomials.
Q: How do I solve a rational equation with multiple variables?
A: To solve a rational equation with multiple variables, you can start by simplifying the equation, if possible. Then, you can use algebraic techniques, such as substitution and elimination, to isolate the variables.
Q: What are some common mistakes to avoid when solving rational equations?
A: Some common mistakes to avoid when solving rational equations include:
- Not simplifying the equation before solving it
- Not canceling out common factors between the numerator and denominator
- Not multiplying both sides of the equation by the LCM
- Not checking for extraneous solutions
Q: How do I check for extraneous solutions?
A: To check for extraneous solutions, you can substitute the solution back into the original equation and check if it is true. If the solution is not true, then it is an extraneous solution and should be discarded.
Q: What are some real-world applications of rational equations?
A: Rational equations have many 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
In this article, we have addressed some of the most frequently asked questions (FAQs) about solving rational equations. We have provided step-by-step explanations and examples to help you better understand the concepts and techniques involved in solving rational equations. By following these tips and avoiding common mistakes, you can become more confident and proficient in solving rational equations.
Final Answer
The final answer to the question "How do I solve a rational equation?" is:
- Simplify the equation, if possible.
- Multiply both sides of the equation by the LCM.
- Use algebraic techniques, such as substitution and elimination, to isolate the variables.
- Check for extraneous solutions.
By following these steps, you can solve rational equations with confidence and accuracy.