Solve For $y$. 6 Y − 5 + Y Y − 3 = 24 Y 2 − 8 Y + 15 \frac{6}{y-5}+\frac{y}{y-3}=\frac{24}{y^2-8y+15} Y − 5 6 ​ + Y − 3 Y ​ = Y 2 − 8 Y + 15 24 ​ If There Is More Than One Solution, Separate Them With Commas. If There Is No Solution, Click On No Solution.

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Introduction

Rational equations are a fundamental concept in algebra, and solving them requires a combination of algebraic techniques and a deep understanding of the underlying mathematics. In this article, we will focus on solving a specific type of rational equation, namely the equation 6y−5+yy−3=24y2−8y+15\frac{6}{y-5}+\frac{y}{y-3}=\frac{24}{y^2-8y+15}. Our goal is to find the value of yy that satisfies this equation, and we will explore different methods and techniques to achieve this.

Understanding the Equation

Before we dive into solving the equation, let's take a closer look at its structure. The equation consists of three fractions, each with a different denominator. The left-hand side of the equation has two fractions, while the right-hand side has one fraction with a quadratic denominator. Our objective is to find a value of yy that makes the equation true.

Step 1: Factor the Denominator

The first step in solving this equation is to factor the denominator of the right-hand side. We can rewrite the denominator as (y−5)(y−3)(y-5)(y-3), which is a product of two binomials. This allows us to rewrite the equation as 6y−5+yy−3=24(y−5)(y−3)\frac{6}{y-5}+\frac{y}{y-3}=\frac{24}{(y-5)(y-3)}.

Step 2: Multiply Both Sides by the Least Common Denominator

To eliminate the fractions, we need to multiply both sides of the equation by the least common denominator (LCD), which is (y−5)(y−3)(y-5)(y-3). This will allow us to simplify the equation and make it easier to solve.

Step 3: Simplify the Equation

After multiplying both sides by the LCD, we get 6(y−3)+y(y−5)=246(y-3)+y(y-5)=24. We can now simplify the equation by expanding and combining like terms.

Step 4: Expand and Combine Like Terms

Expanding the left-hand side of the equation, we get 6y−18+yy−5y=246y-18+yy-5y=24. Combining like terms, we get 7y−18=247y-18=24.

Step 5: Solve for y

To solve for yy, we need to isolate the variable on one side of the equation. We can do this by adding 18 to both sides of the equation, which gives us 7y=427y=42. Dividing both sides by 7, we get y=6y=6.

Conclusion

In this article, we have solved the rational equation 6y−5+yy−3=24y2−8y+15\frac{6}{y-5}+\frac{y}{y-3}=\frac{24}{y^2-8y+15}. We have used a combination of algebraic techniques, including factoring, multiplying by the LCD, and simplifying the equation. Our final answer is y=6y=6, which is the only solution to the equation.

Discussion

Rational equations are a fundamental concept in algebra, and solving them requires a combination of algebraic techniques and a deep understanding of the underlying mathematics. In this article, we have focused on solving a specific type of rational equation, namely the equation 6y−5+yy−3=24y2−8y+15\frac{6}{y-5}+\frac{y}{y-3}=\frac{24}{y^2-8y+15}. Our goal is to find the value of yy that satisfies this equation, and we have explored different methods and techniques to achieve this.

Tips and Tricks

When solving rational equations, it's essential to remember the following tips and tricks:

  • Factor the denominator whenever possible.
  • Multiply both sides of the equation by the LCD to eliminate the fractions.
  • Simplify the equation by expanding and combining like terms.
  • Isolate the variable on one side of the equation to solve for yy.

Common Mistakes

When solving rational equations, it's easy to make mistakes. Here are some common mistakes to avoid:

  • Failing to factor the denominator.
  • Multiplying both sides of the equation by the wrong LCD.
  • Failing to simplify the equation.
  • Not isolating the variable on one side of the equation.

Real-World Applications

Rational equations have numerous real-world applications in fields such as engineering, physics, and economics. For example, rational equations can be used to model population growth, electrical circuits, and financial markets.

Conclusion

In conclusion, solving rational equations requires a combination of algebraic techniques and a deep understanding of the underlying mathematics. By following the steps outlined in this article, you can solve rational equations with confidence. Remember to factor the denominator, multiply both sides by the LCD, simplify the equation, and isolate the variable on one side of the equation. With practice and patience, you can become proficient in solving rational equations and apply them to real-world problems.

Final Answer

The final answer is: 6\boxed{6}

Introduction

Rational equations are a fundamental concept in algebra, and solving them requires a combination of algebraic techniques and a deep understanding of the underlying mathematics. In this article, we will answer some of the most frequently asked questions about rational equations, providing you with a comprehensive guide to solving these types of equations.

Q1: What is a rational equation?

A1: A rational equation is an equation that contains one or more rational expressions, which are expressions that can be written as the ratio of two polynomials.

Q2: How do I solve a rational equation?

A2: To solve a rational equation, you need to follow these steps:

  1. Factor the denominator of the equation, if possible.
  2. Multiply both sides of the equation by the least common denominator (LCD) to eliminate the fractions.
  3. Simplify the equation by expanding and combining like terms.
  4. Isolate the variable on one side of the equation to solve for y.

Q3: What is the least common denominator (LCD)?

A3: The least common denominator (LCD) is the smallest expression that can be divided evenly by all the denominators in the equation.

Q4: How do I find the LCD?

A4: To find the LCD, you need to factor all the denominators in the equation and then multiply the factors together.

Q5: What if the equation has no solution?

A5: If the equation has no solution, it means that the equation is inconsistent, and there is no value of y that can satisfy the equation.

Q6: What if the equation has multiple solutions?

A6: If the equation has multiple solutions, it means that there are multiple values of y that can satisfy the equation.

Q7: Can I use a calculator to solve rational equations?

A7: Yes, you can use a calculator to solve rational equations, but it's essential to understand the underlying mathematics and the steps involved in solving the equation.

Q8: How do I check my answer?

A8: To check your answer, you need to plug the value of y back into the original equation and verify that it is true.

Q9: What are some common mistakes to avoid when solving rational equations?

A9: Some common mistakes to avoid when solving rational equations include:

  • Failing to factor the denominator.
  • Multiplying both sides of the equation by the wrong LCD.
  • Failing to simplify the equation.
  • Not isolating the variable on one side of the equation.

Q10: How can I practice solving rational equations?

A10: You can practice solving rational equations by working through examples and exercises in your textbook or online resources. You can also try solving rational equations on your own and then check your answers with a calculator or a teacher.

Conclusion

In conclusion, rational equations are a fundamental concept in algebra, and solving them requires a combination of algebraic techniques and a deep understanding of the underlying mathematics. By following the steps outlined in this article and practicing regularly, you can become proficient in solving rational equations and apply them to real-world problems.

Final Answer

The final answer is: 6\boxed{6}