Solve The Given Equation. Check Your Solution. Write Your Answer In Ascending Order.$\[ \frac{-12}{y} = Y - 7 \\]$\[ y = \square, \square \\]

Introduction

In this article, we will delve into solving a given equation, which involves isolating the variable y. The equation is $\frac{-12}{y} = y - 7$. Our goal is to find the values of y that satisfy this equation and present the solutions in ascending order.

Step 1: Multiply Both Sides by y

To begin solving the equation, we need to eliminate the fraction. We can do this by multiplying both sides of the equation by y. This will give us:

$$-12 = y^2 - 7y$$

Step 2: Rearrange the Equation

Next, we need to rearrange the equation to form a quadratic equation. We can do this by moving all terms to one side of the equation:

$$y^2 - 7y + 12 = 0$$

Step 3: Factor the Quadratic Equation

Now, we need to factor the quadratic equation. We can do this by finding two numbers whose product is 12 and whose sum is -7. These numbers are -3 and -4, so we can factor the equation as:

$$(y - 3)(y - 4) = 0$$

Step 4: Solve for y

To find the values of y, we need to set each factor equal to zero and solve for y. This gives us:

$$y - 3 = 0 \Rightarrow y = 3$$

$$y - 4 = 0 \Rightarrow y = 4$$

Step 5: Check the Solutions

Now that we have found the values of y, we need to check if they satisfy the original equation. We can do this by plugging each value of y back into the original equation:

For y = 3:

$$\frac{-12}{3} = 3 - 7 \Rightarrow -4 = -4$$

For y = 4:

$$\frac{-12}{4} = 4 - 7 \Rightarrow -3 = -3$$

Both values of y satisfy the original equation, so we can be confident that our solutions are correct.

Conclusion

In this article, we have solved the given equation $\frac{-12}{y} = y - 7$ and presented the solutions in ascending order. We have followed a step-by-step approach to isolate the variable y and have checked our solutions to ensure that they satisfy the original equation. The solutions are y = 3 and y = 4.

Final Answer

The final answer is $\boxed{3, 4}$.

Discussion

This equation is a quadratic equation, which can be solved using various methods such as factoring, quadratic formula, or graphing. In this article, we have used the factoring method to solve the equation. The solutions to the equation are the values of y that satisfy the equation, and they can be found by setting each factor equal to zero and solving for y.

Related Topics

• Solving quadratic equations
• Factoring quadratic equations
• Quadratic formula
• Graphing quadratic equations

References

• [1] "Algebra" by Michael Artin
• [2] "Calculus" by Michael Spivak
• [3] "Mathematics for Computer Science" by Eric Lehman and Tom Leighton
  Solving the Given Equation: A Q&A Article =====================================================

Introduction

In our previous article, we solved the given equation $\frac{-12}{y} = y - 7$ and presented the solutions in ascending order. In this article, we will answer some frequently asked questions related to the equation and its solutions.

Q: What is the equation $\frac{-12}{y} = y - 7$?

A: The equation $\frac{-12}{y} = y - 7$ is a quadratic equation that involves the variable y. It is a type of equation that can be solved using various methods such as factoring, quadratic formula, or graphing.

Q: How do I solve the equation $\frac{-12}{y} = y - 7$?

A: To solve the equation $\frac{-12}{y} = y - 7$, you can follow these steps:

1. Multiply both sides of the equation by y to eliminate the fraction.
2. Rearrange the equation to form a quadratic equation.
3. Factor the quadratic equation.
4. Set each factor equal to zero and solve for y.

Q: What are the solutions to the equation $\frac{-12}{y} = y - 7$?

A: The solutions to the equation $\frac{-12}{y} = y - 7$ are y = 3 and y = 4.

Q: How do I check if the solutions satisfy the original equation?

A: To check if the solutions satisfy the original equation, you can plug each value of y back into the original equation and verify that it is true.

Q: What is the significance of the solutions y = 3 and y = 4?

A: The solutions y = 3 and y = 4 are the values of y that satisfy the equation $\frac{-12}{y} = y - 7$. They represent the points where the graph of the equation intersects the y-axis.

Q: Can I use other methods to solve the equation $\frac{-12}{y} = y - 7$?

A: Yes, you can use other methods to solve the equation $\frac{-12}{y} = y - 7$, such as the quadratic formula or graphing. However, the factoring method is a simple and efficient way to solve the equation.

Q: What are some real-world applications of solving quadratic equations?

A: Solving quadratic equations has many real-world applications, such as:

• Modeling population growth or decline
• Determining the maximum or minimum value of a function
• Finding the intersection points of two curves
• Solving optimization problems

Conclusion

In this article, we have answered some frequently asked questions related to the equation $\frac{-12}{y} = y - 7$ and its solutions. We hope that this article has provided you with a better understanding of the equation and its significance.

Final Answer

The final answer is $\boxed{3, 4}$.

Discussion

This equation is a quadratic equation, which can be solved using various methods such as factoring, quadratic formula, or graphing. In this article, we have used the factoring method to solve the equation. The solutions to the equation are the values of y that satisfy the equation, and they can be found by setting each factor equal to zero and solving for y.

Related Topics

• Solving quadratic equations
• Factoring quadratic equations
• Quadratic formula
• Graphing quadratic equations

References

• [1] "Algebra" by Michael Artin
• [2] "Calculus" by Michael Spivak
• [3] "Mathematics for Computer Science" by Eric Lehman and Tom Leighton