Solve For $y$.$\frac{12}{y^2-3y-4}=-\frac{4y}{y+1}$If There Is More Than One Solution, Separate Them With Commas. If There Is No Solution, Select No Solution.

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Understanding the Problem

To solve the given equation, we need to isolate the variable yy. The equation involves fractions, so we will start by eliminating the fractions to simplify the equation. We will then use algebraic techniques to solve for yy.

Step 1: Eliminate the Fractions

To eliminate the fractions, we can multiply both sides of the equation by the least common multiple (LCM) of the denominators. In this case, the LCM is (y2−3y−4)(y+1)(y^2-3y-4)(y+1).

\frac{12}{y^2-3y-4}=-\frac{4y}{y+1}

Multiplying both sides by (y2−3y−4)(y+1)(y^2-3y-4)(y+1), we get:

12(y+1)=-4y(y^2-3y-4)

Step 2: Expand and Simplify

Expanding the right-hand side of the equation, we get:

12y+12=-4y^3+12y^2+16y

Rearranging the terms, we get:

-4y^3+12y^2+28y+12=0

Step 3: Factor the Equation

We can factor the equation by grouping the terms:

-4y^3+12y^2+28y+12=-4y(y^2-3y-3)-4(y^2-3y-3)=0

Factoring the quadratic expression, we get:

-4y(y-6)(y+1)-4(y-6)(y+1)=0

Step 4: Solve for yy

We can now solve for yy by setting each factor equal to zero:

-4y(y-6)(y+1)=0

This gives us three possible solutions:

y=0, y=6, y=-1

However, we need to check if these solutions are valid by plugging them back into the original equation.

Step 5: Check the Solutions

Plugging y=0y=0 into the original equation, we get:

\frac{12}{-4}=-\frac{0}{1}

This is a true statement, so y=0y=0 is a valid solution.

Plugging y=6y=6 into the original equation, we get:

\frac{12}{-4}=-\frac{24}{7}

This is a false statement, so y=6y=6 is not a valid solution.

Plugging y=−1y=-1 into the original equation, we get:

\frac{12}{-4}=-\frac{-4}{0}

This is a false statement, so y=−1y=-1 is not a valid solution.

Conclusion

The only valid solution to the equation is y=0y=0.

Final Answer

The final answer is 0\boxed{0}.

Introduction

Solving equations with fractions can be a challenging task, but with the right techniques and strategies, it can be made easier. In this article, we will provide a step-by-step guide on how to solve equations with fractions, along with some frequently asked questions and answers.

Q: What is the first step in solving an equation with fractions?

A: The first step in solving an equation with fractions is to eliminate the fractions by multiplying both sides of the equation by the least common multiple (LCM) of the denominators.

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. Alternatively, you can use the formula: LCM(a, b) = (a × b) / GCD(a, b), where GCD is the greatest common divisor.

Q: What if the LCM is a complex expression?

A: If the LCM is a complex expression, you can simplify it by factoring out common terms. For example, if the LCM is (x^2 + 4x + 4), you can factor it as (x + 2)^2.

Q: How do I expand and simplify the equation after multiplying by the LCM?

A: After multiplying both sides of the equation by the LCM, you need to expand and simplify the equation by combining like terms. This may involve distributing the LCM to each term in the equation.

Q: What if the equation has multiple solutions?

A: If the equation has multiple solutions, you need to check each solution by plugging it back into the original equation. This will help you determine which solutions are valid and which ones are not.

Q: How do I check if a solution is valid?

A: To check if a solution is valid, you need to plug it back into the original equation and simplify. If the resulting equation is true, then the solution is valid. If the resulting equation is false, then the solution is not valid.

Q: What if I get a false statement when plugging in a solution?

A: If you get a false statement when plugging in a solution, then the solution is not valid. This means that the solution does not satisfy the original equation, and therefore, it is not a valid solution.

Q: Can I use algebraic techniques to solve equations with fractions?

A: Yes, you can use algebraic techniques to solve equations with fractions. In fact, algebraic techniques are often the most effective way to solve equations with fractions.

Q: What are some common algebraic techniques used to solve equations with fractions?

A: Some common algebraic techniques used to solve equations with fractions include factoring, expanding, and simplifying. You can also use techniques such as substitution and elimination to solve equations with fractions.

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

A: To determine if an equation has a solution, you need to check if the equation is true for any value of the variable. If the equation is true for any value of the variable, then the equation has a solution.

Q: What if an equation has no solution?

A: If an equation has no solution, then the equation is false for all values of the variable. This means that there is no value of the variable that can make the equation true.

Conclusion

Solving equations with fractions can be a challenging task, but with the right techniques and strategies, it can be made easier. By following the steps outlined in this article, you can learn how to solve equations with fractions and become more confident in your ability to solve mathematical problems.

Final Answer

The final answer is 0\boxed{0}.