Solve The Equation Y -1 Y-2 = 1. 3 4​

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Introduction

In this article, we will delve into the world of algebra and focus on solving a specific equation. The equation in question is y - 1 / y - 2 = 1 / 3 / 4. This equation may seem complex at first glance, but with the right approach and techniques, we can simplify it and find the solution. We will break down the equation step by step, and by the end of this article, you will have a clear understanding of how to solve it.

Understanding the Equation

Before we dive into solving the equation, let's take a closer look at it. The equation is y - 1 / y - 2 = 1 / 3 / 4. This equation involves fractions and variables, which can make it challenging to solve. However, with the right approach, we can simplify it and find the solution.

Simplifying the Equation

To simplify the equation, we need to get rid of the fractions. We can do this by multiplying both sides of the equation by the least common multiple (LCM) of the denominators. In this case, the LCM of 3 and 4 is 12. So, we multiply both sides of the equation by 12.

12 * (y - 1 / y - 2) = 12 * (1 / 3 / 4)

This simplifies the equation to:

12y - 12 / y - 24 = 3

Eliminating the Fractions

Now that we have simplified the equation, we can eliminate the fractions. We can do this by multiplying both sides of the equation by the denominator, which is y - 24.

(12y - 12) * (y - 24) = 3 * (y - 24)

This simplifies the equation to:

12y^2 - 288y - 12y + 288 = 3y - 72

Combining Like Terms

Now that we have eliminated the fractions, we can combine like terms. We can do this by combining the terms with the same variable.

12y^2 - 300y + 288 = 3y - 72

Rearranging the Equation

Now that we have combined like terms, we can rearrange the equation to get all the terms on one side. We can do this by subtracting 3y from both sides of the equation and adding 72 to both sides.

12y^2 - 303y + 360 = 0

Solving the Quadratic Equation

Now that we have rearranged the equation, we can solve it using the quadratic formula. The quadratic formula is:

y = (-b ± √(b^2 - 4ac)) / 2a

In this case, a = 12, b = -303, and c = 360. Plugging these values into the quadratic formula, we get:

y = (303 ± √((-303)^2 - 4 * 12 * 360)) / (2 * 12)

Simplifying this expression, we get:

y = (303 ± √(91809 - 17280)) / 24

This simplifies to:

y = (303 ± √74529) / 24

Finding the Solutions

Now that we have the quadratic formula, we can find the solutions to the equation. We can do this by plugging in the values of a, b, and c into the quadratic formula and simplifying the expression.

y = (303 ± √74529) / 24

Simplifying this expression, we get:

y = (303 ± 272.5) / 24

This gives us two possible solutions:

y = (303 + 272.5) / 24
y = (303 - 272.5) / 24

Simplifying these expressions, we get:

y = 575.5 / 24
y = 30.5 / 24

This gives us two possible solutions:

y = 24.0
y = 1.25

Conclusion

In this article, we have solved the equation y - 1 / y - 2 = 1 / 3 / 4. We started by simplifying the equation and eliminating the fractions. We then combined like terms and rearranged the equation to get all the terms on one side. Finally, we solved the quadratic equation using the quadratic formula and found two possible solutions. These solutions are y = 24.0 and y = 1.25.

Final Answer

The final answer is y = 24.0 and y = 1.25.

Introduction

In our previous article, we solved the equation y - 1 / y - 2 = 1 / 3 / 4. We broke down the equation step by step, simplified it, and found two possible solutions. In this article, we will answer some of the most frequently asked questions about solving this equation.

Q: What is the first step in solving the equation y - 1 / y - 2 = 1 / 3 / 4?

A: The first step in solving the equation is to simplify it by getting rid of the fractions. We can do this by multiplying both sides of the equation by the least common multiple (LCM) of the denominators, which is 12.

Q: How do I eliminate the fractions in the equation?

A: To eliminate the fractions, we can multiply both sides of the equation by the denominator, which is y - 24. This will get rid of the fractions and make it easier to solve the equation.

Q: What is the quadratic formula, and how do I use it to solve the equation?

A: The quadratic formula is a mathematical formula that is used to solve quadratic equations. It is given by:

y = (-b ± √(b^2 - 4ac)) / 2a

To use the quadratic formula, we need to plug in the values of a, b, and c into the formula and simplify the expression.

Q: What are the two possible solutions to the equation?

A: The two possible solutions to the equation are y = 24.0 and y = 1.25. These solutions were found by plugging in the values of a, b, and c into the quadratic formula and simplifying the expression.

Q: How do I check my solutions to make sure they are correct?

A: To check your solutions, you can plug them back into the original equation and see if they are true. If they are true, then you have found the correct solutions.

Q: What if I get stuck while solving the equation?

A: If you get stuck while solving the equation, don't worry! You can try breaking down the equation into smaller steps, or you can ask for help from a teacher or tutor.

Q: Can I use a calculator to solve the equation?

A: Yes, you can use a calculator to solve the equation. However, keep in mind that using a calculator may not help you understand the underlying math, so it's still a good idea to work through the problem by hand.

Q: How long does it take to solve the equation?

A: The time it takes to solve the equation will depend on your level of math proficiency and your familiarity with the quadratic formula. With practice, you should be able to solve the equation in a few minutes.

Q: Can I use the quadratic formula to solve other types of equations?

A: Yes, the quadratic formula can be used to solve other types of equations, such as quadratic equations with complex coefficients.

Q: What are some common mistakes to avoid when solving the equation?

A: Some common mistakes to avoid when solving the equation include:

  • Not simplifying the equation enough before trying to solve it
  • Not using the correct values for a, b, and c in the quadratic formula
  • Not checking your solutions to make sure they are correct

Conclusion

In this article, we have answered some of the most frequently asked questions about solving the equation y - 1 / y - 2 = 1 / 3 / 4. We have covered topics such as simplifying the equation, eliminating fractions, using the quadratic formula, and checking solutions. We hope that this article has been helpful in answering your questions and providing you with a better understanding of how to solve this equation.

Final Answer

The final answer is y = 24.0 and y = 1.25.