Given That { X $}$ Varies Inversely With { Y $}$ And { X Y = 2 $}$, What Is The Value Of { X $}$ When { Y = 1 $}$?A. { -1$}$B. { \frac{1}{2}$}$C. 1D. 2

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. This relationship is often represented by the equation:

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

where $x$ and $y$ are the variables, and $k$ is a constant.

Given Information

We are given that $x$ varies inversely with $y$, and the product of $x$ and $y$ is equal to 2, i.e., $xy = 2$. We need to find the value of $x$ when $y = 1$.

Step 1: Write the Equation of Inverse Variation

Since $x$ varies inversely with $y$, we can write the equation as:

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

Step 2: Use the Given Information to Find the Value of $k$

We are given that $xy = 2$. Substituting this into the equation, we get:

$$x \cdot y = 2$$

$$\frac{k}{y} \cdot y = 2$$

$$k = 2$$

Step 3: Find the Value of $x$ When $y = 1$

Now that we have the value of $k$, we can substitute $y = 1$ into the equation to find the value of $x$:

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

$$x = \frac{2}{1}$$

$$x = 2$$

Conclusion

Therefore, the value of $x$ when $y = 1$ is 2.

Answer

The correct answer is D. 2.

Discussion

Inverse variation is a powerful tool in mathematics that helps us understand the relationship between two variables. By using the given information and the equation of inverse variation, we can find the value of $x$ when $y = 1$. This problem requires a clear understanding of inverse variation and its applications.

Real-World Applications

Inverse variation has many real-world applications, such as:

• Physics: Inverse variation is used to describe the relationship between the force of gravity and the distance between two objects.
• Economics: Inverse variation is used to describe the relationship between the price of a commodity and its demand.
• Biology: Inverse variation is used to describe the relationship between the concentration of a substance and its rate of diffusion.

Tips and Tricks

• Pay attention to the units: When working with inverse variation, make sure to pay attention to the units of the variables.
• Use the given information: Use the given information to find the value of the constant $k$.
• Substitute the values: Substitute the values of the variables into the equation to find the solution.

Practice Problems

• Problem 1: If $x$ varies inversely with $y$, and $xy = 5$, find the value of $x$ when $y = 2$.
• Problem 2: If $x$ varies inversely with $y$, and $xy = 10$, find the value of $x$ when $y = 5$.

Conclusion

$$$$

Frequently Asked Questions

Q: What is inverse variation?

A: Inverse variation is a relationship between two variables where one variable increases as the other decreases, and vice versa. It is often represented by the equation:

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

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

A: Direct variation is a relationship where one variable increases as the other increases, and vice versa. Inverse variation is a relationship where one variable increases as the other decreases, and vice versa.

Q: How do I determine if a relationship is inverse variation?

A: To determine if a relationship is inverse variation, look for the following characteristics:

• The variables are related in a way that one increases as the other decreases.
• The relationship can be represented by the equation:

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

Q: What is the constant $k$ in the equation of inverse variation?

A: The constant $k$ is a value that is multiplied by the reciprocal of the variable $y$ to get the value of $x$. It is a measure of the strength of the relationship between the variables.

Q: How do I find the value of $k$ in the equation of inverse variation?

A: To find the value of $k$, use the given information to set up an equation. For example, if $xy = 2$, you can set up the equation:

$$x \cdot y = 2$$

$$\frac{k}{y} \cdot y = 2$$

$$k = 2$$

Q: What is the relationship between the variables in an inverse variation?

A: In an inverse variation, the variables are related in a way that one increases as the other decreases. This means that as one variable gets larger, the other variable gets smaller, and vice versa.

Q: How do I use the equation of inverse variation to solve problems?

A: To use the equation of inverse variation to solve problems, follow these steps:

1. Write the equation of inverse variation:

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

2. Use the given information to find the value of $k$.

3. Substitute the values of the variables into the equation to find the solution.

Q: What are some real-world applications of inverse variation?

A: Inverse variation has many real-world applications, such as:

• Physics: Inverse variation is used to describe the relationship between the force of gravity and the distance between two objects.
• Economics: Inverse variation is used to describe the relationship between the price of a commodity and its demand.
• Biology: Inverse variation is used to describe the relationship between the concentration of a substance and its rate of diffusion.

Q: What are some tips and tricks for working with inverse variation?

A: Here are some tips and tricks for working with inverse variation:

• Pay attention to the units: When working with inverse variation, make sure to pay attention to the units of the variables.
• Use the given information: Use the given information to find the value of the constant $k$.
• Substitute the values: Substitute the values of the variables into the equation to find the solution.

Q: What are some practice problems for inverse variation?

A: Here are some practice problems for inverse variation:

• Problem 1: If $x$ varies inversely with $y$, and $xy = 5$, find the value of $x$ when $y = 2$.
• Problem 2: If $x$ varies inversely with $y$, and $xy = 10$, find the value of $x$ when $y = 5$.

Conclusion

In conclusion, inverse variation is a fundamental concept in mathematics that describes the relationship between two variables. By understanding the equation of inverse variation and how to use it to solve problems, you can apply this concept to a wide range of real-world applications.