If { Y $}$ Varies Inversely With { X $}$, And { Y = -5 $}$ When { X = 20 $}$, Find { X $}$ When { Y = -4 $}$.A. 16 B. 25 C. -100 D. -25

Understanding Inverse Variation

Inverse variation is a fundamental concept in mathematics that describes the relationship between two variables. It states that as one variable increases, the other variable decreases, and vice versa. In this article, we will explore the concept of inverse variation, its formula, and how to apply it to solve problems.

What is Inverse Variation?

Inverse variation is a type of functional relationship where the product of two variables remains constant. Mathematically, it can be represented as:

$$y = \frac{k}{x}$$

where $y$ is the dependent variable, $x$ is the independent variable, and $k$ is a constant.

Example: Inverse Variation in Real-Life Scenarios

Inverse variation can be observed in various real-life scenarios, such as:

• The distance between two objects and the force of attraction between them
• The amount of time it takes to complete a task and the number of workers involved
• The cost of a product and the quantity demanded

The Formula for Inverse Variation

The formula for inverse variation is:

$$y = \frac{k}{x}$$

where $y$ is the dependent variable, $x$ is the independent variable, and $k$ is a constant.

Solving Inverse Variation Problems

To solve inverse variation problems, we need to follow these steps:

1. Write the equation of inverse variation using the given information.
2. Substitute the values of $y$ and $x$ into the equation.
3. Solve for the constant $k$.
4. Use the value of $k$ to find the value of $x$ or $y$.

Example Problem: If { y $}$ varies inversely with { x $}$, and { y = -5 $}$ when { x = 20 $}$, find { x $}$ when { y = -4 $}$.

Let's use the formula for inverse variation to solve this problem.

Step 1: Write the equation of inverse variation using the given information.

$$y = \frac{k}{x}$$

Step 2: Substitute the values of $y$ and $x$ into the equation.

$$-5 = \frac{k}{20}$$

Step 3: Solve for the constant $k$.

$$k = -5 \times 20$$

$$k = -100$$

Step 4: Use the value of $k$ to find the value of $x$ when $y = -4$.

$$-4 = \frac{-100}{x}$$

$$x = \frac{-100}{-4}$$

$$x = 25$$

Therefore, the value of $x$ when $y = -4$ is 25.

Conclusion

Inverse variation is a fundamental concept in mathematics that describes the relationship between two variables. It states that as one variable increases, the other variable decreases, and vice versa. In this article, we have explored the concept of inverse variation, its formula, and how to apply it to solve problems. We have also solved an example problem to demonstrate how to use the formula for inverse variation.

Key Takeaways

• Inverse variation is a type of functional relationship where the product of two variables remains constant.
• The formula for inverse variation is $y = \frac{k}{x}$, where $y$ is the dependent variable, $x$ is the independent variable, and $k$ is a constant.
• To solve inverse variation problems, we need to follow the steps: write the equation of inverse variation, substitute the values of $y$ and $x$ into the equation, solve for the constant $k$, and use the value of $k$ to find the value of $x$ or $y$.

Practice Problems

1. If $y$ varies inversely with $x$, and $y = 3$ when $x = 4$, find $x$ when $y = 2$.
2. If $y$ varies inversely with $x$, and $y = -2$ when $x = 6$, find $y$ when $x = 3$.
3. If $y$ varies inversely with $x$, and $y = 5$ when $x = 8$, find $x$ when $y = 3$.

Answer Key

1. $x = 8$
2. $y = -1$
3. $x = 10$

References

• [1] Khan Academy. (n.d.). Inverse Variation. Retrieved from https://www.khanacademy.org/math/algebra/x2f1f4/inverse-variation
• [2] Math Open Reference. (n.d.). Inverse Variation. Retrieved from https://www.mathopenref.com/inversevariation.html
• [3] Purplemath. (n.d.). Inverse Variation. Retrieved from https://www.purplemath.com/modules/inversevar.htm
  Inverse Variation Q&A =========================

Frequently Asked Questions

Q: What is inverse variation?

A: Inverse variation is a type of functional relationship where the product of two variables remains constant. Mathematically, it can be represented as:

$$y = \frac{k}{x}$$

where $y$ is the dependent variable, $x$ is the independent variable, and $k$ is a constant.

Q: What is the formula for inverse variation?

A: The formula for inverse variation is:

$$y = \frac{k}{x}$$

where $y$ is the dependent variable, $x$ is the independent variable, and $k$ is a constant.

Q: How do I solve inverse variation problems?

A: To solve inverse variation problems, you need to follow these steps:

1. Write the equation of inverse variation using the given information.
2. Substitute the values of $y$ and $x$ into the equation.
3. Solve for the constant $k$.
4. Use the value of $k$ to find the value of $x$ or $y$.

Q: What is the difference between inverse variation and direct variation?

A: Inverse variation and direct variation are two types of functional relationships. Direct variation is a relationship where the product of two variables remains constant, while inverse variation is a relationship where the product of two variables remains constant, but the variables are in opposite directions.

Q: Can you give an example of inverse variation in real-life scenarios?

A: Yes, inverse variation can be observed in various real-life scenarios, such as:

• The distance between two objects and the force of attraction between them
• The amount of time it takes to complete a task and the number of workers involved
• The cost of a product and the quantity demanded

Q: How do I determine the value of the constant $k$ in an inverse variation problem?

A: To determine the value of the constant $k$ in an inverse variation problem, you need to substitute the values of $y$ and $x$ into the equation and solve for $k$.

Q: Can you give an example of how to solve an inverse variation problem?

A: Yes, let's use the following example:

If $y$ varies inversely with $x$, and $y = -5$ when $x = 20$, find $x$ when $y = -4$.

Step 1: Write the equation of inverse variation using the given information.

$$y = \frac{k}{x}$$

Step 2: Substitute the values of $y$ and $x$ into the equation.

$$-5 = \frac{k}{20}$$

Step 3: Solve for the constant $k$.

$$k = -5 \times 20$$

$$k = -100$$

Step 4: Use the value of $k$ to find the value of $x$ when $y = -4$.

$$-4 = \frac{-100}{x}$$

$$x = \frac{-100}{-4}$$

$$x = 25$$

Therefore, the value of $x$ when $y = -4$ is 25.

Q: What are some common mistakes to avoid when solving inverse variation problems?

A: Some common mistakes to avoid when solving inverse variation problems include:

• Not writing the equation of inverse variation correctly
• Not substituting the values of $y$ and $x$ into the equation correctly
• Not solving for the constant $k$ correctly
• Not using the value of $k$ to find the value of $x$ or $y$ correctly

Q: Can you give some practice problems to help me understand inverse variation better?

A: Yes, here are some practice problems to help you understand inverse variation better:

1. If $y$ varies inversely with $x$, and $y = 3$ when $x = 4$, find $x$ when $y = 2$.
2. If $y$ varies inversely with $x$, and $y = -2$ when $x = 6$, find $y$ when $x = 3$.
3. If $y$ varies inversely with $x$, and $y = 5$ when $x = 8$, find $x$ when $y = 3$.

Answer Key

1. $x = 8$
2. $y = -1$
3. $x = 10$

References

• [1] Khan Academy. (n.d.). Inverse Variation. Retrieved from https://www.khanacademy.org/math/algebra/x2f1f4/inverse-variation
• [2] Math Open Reference. (n.d.). Inverse Variation. Retrieved from https://www.mathopenref.com/inversevariation.html
• [3] Purplemath. (n.d.). Inverse Variation. Retrieved from https://www.purplemath.com/modules/inversevar.htm