Suppose $y$ Varies Inversely With $x$. If \$y$[/tex\] Is 5 When $x$ Is 14, What Is $y$ When \$x$[/tex\] Is 7?A. $y = 2.5$ B. $y = 10$ C. $y = -2$ D.

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 delve into the world of inverse variation and explore its applications in various fields.

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 force of gravity between two objects decreases as the distance between them increases.
• The pressure of a gas decreases as the volume of the container increases.
• The time it takes to complete a task decreases as the number of workers increases.

Solving Inverse Variation Problems

To solve inverse variation problems, we need to follow a step-by-step approach:

1. Identify the variables: Determine the dependent and independent variables in the problem.
2. Write the equation: Write the inverse variation equation using the variables and the constant $k$.
3. Plug in the values: Plug in the given values into the equation.
4. Solve for the unknown: Solve for the unknown variable.

Suppose $y$ varies inversely with $x$. If $$y$[/tex] is 5 when $x$ is 14, what is $y$ when $$x$[/tex] is 7?

To solve this problem, we need to follow the step-by-step approach outlined above.

1. Identify the variables: The dependent variable is $y$, and the independent variable is $x$.
2. Write the equation: The inverse variation equation is:

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

3. Plug in the values: We are given that $y$ is 5 when $x$ is 14. Plugging in these values, we get:

$$5 = \frac{k}{14}$$

4. Solve for the unknown: To solve for $k$, we can multiply both sides of the equation by 14:

$$k = 5 \times 14$$

$$k = 70$$

Now that we have found the value of $k$, we can plug in the new value of $x$ (7) into the equation to find the value of $y$:

$$y = \frac{70}{7}$$

$$y = 10$$

Therefore, the value of $y$ when $x$ is 7 is 10.

Conclusion

Inverse variation is a fundamental concept in mathematics that describes the relationship between two variables. It has numerous applications in various fields, including physics, engineering, and economics. By understanding inverse variation, we can solve problems that involve the relationship between two variables. In this article, we have explored the concept of inverse variation and provided a step-by-step approach to solving inverse variation problems.

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.

Q: How do I solve inverse variation problems?

A: To solve inverse variation problems, you need to follow a step-by-step approach: identify the variables, write the equation, plug in the values, and solve for the unknown.

Q: What is the formula for inverse variation?

A: The formula for inverse variation is:

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

Q: What is the value of $y$ when $x$ is 7?

A: The value of $y$ when $x$ is 7 is 10.

References

• Khan Academy. (n.d.). Inverse Variation. Retrieved from https://www.khanacademy.org/math/algebra/x2f1f4f/inverse-variation
• Math Open Reference. (n.d.). Inverse Variation. Retrieved from https://www.mathopenref.com/inversevariation.html
• Wolfram MathWorld. (n.d.). Inverse Variation. Retrieved from https://mathworld.wolfram.com/InverseVariation.html

Further Reading

• Algebra: A Comprehensive Introduction by Michael Artin
• Calculus: Early Transcendentals by James Stewart
• Mathematics for Computer Science by Eric Lehman, F Thomson Leighton, and Albert R Meyer
  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. It can be represented mathematically as:

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

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 a step-by-step approach:

1. Identify the variables: Determine the dependent and independent variables in the problem.
2. Write the equation: Write the inverse variation equation using the variables and the constant $k$.
3. Plug in the values: Plug in the given values into the equation.
4. Solve for the unknown: Solve for the unknown variable.

Q: What is the formula for inverse variation?

A: The formula for inverse variation is:

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

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, whereas inverse variation is a relationship where the product of two variables remains constant, but the variables are in opposite directions.

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

A: Yes, here are a few examples of inverse variation in real-life scenarios:

• The force of gravity between two objects decreases as the distance between them increases.
• The pressure of a gas decreases as the volume of the container increases.
• The time it takes to complete a task decreases as the number of workers increases.

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

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

Q: Can you provide a step-by-step solution to an inverse variation problem?

A: Yes, here is a step-by-step solution to an inverse variation problem:

Suppose $y$ varies inversely with $x$. If $y$ is 5 when $x$ is 14, what is $y$ when $x$ is 7?

1. Identify the variables: The dependent variable is $y$, and the independent variable is $x$.
2. Write the equation: The inverse variation equation is:

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

3. Plug in the values: We are given that $y$ is 5 when $x$ is 14. Plugging in these values, we get:

$$5 = \frac{k}{14}$$

4. Solve for the unknown: To solve for $k$, we can multiply both sides of the equation by 14:

$$k = 5 \times 14$$

$$k = 70$$

Now that we have found the value of $k$, we can plug in the new value of $x$ (7) into the equation to find the value of $y$:

$$y = \frac{70}{7}$$

$$y = 10$$

Therefore, the value of $y$ when $x$ is 7 is 10.

Q: Can you provide more examples of inverse variation problems?

A: Yes, here are a few more examples of inverse variation problems:

• Suppose $y$ varies inversely with $x$. If $y$ is 10 when $x$ is 2, what is $y$ when $x$ is 5?
• Suppose $y$ varies inversely with $x$. If $y$ is 20 when $x$ is 3, what is $y$ when $x$ is 6?
• Suppose $y$ varies inversely with $x$. If $y$ is 15 when $x$ is 4, what is $y$ when $x$ is 9?

Conclusion

Inverse variation is a fundamental concept in mathematics that describes the relationship between two variables. It has numerous applications in various fields, including physics, engineering, and economics. By understanding inverse variation, we can solve problems that involve the relationship between two variables. In this article, we have explored the concept of inverse variation and provided a step-by-step approach to solving inverse variation problems.

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.

Q: How do I solve inverse variation problems?

A: To solve inverse variation problems, you need to follow a step-by-step approach: identify the variables, write the equation, plug in the values, and solve for the unknown.

Q: What is the formula for inverse variation?

A: The formula for inverse variation is:

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

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, whereas inverse variation is a relationship where the product of two variables remains constant, but the variables are in opposite directions.

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

A: Yes, here are a few examples of inverse variation in real-life scenarios:

• The force of gravity between two objects decreases as the distance between them increases.
• The pressure of a gas decreases as the volume of the container increases.
• The time it takes to complete a task decreases as the number of workers increases.

References

• Khan Academy. (n.d.). Inverse Variation. Retrieved from https://www.khanacademy.org/math/algebra/x2f1f4f/inverse-variation
• Math Open Reference. (n.d.). Inverse Variation. Retrieved from https://www.mathopenref.com/inversevariation.html
• Wolfram MathWorld. (n.d.). Inverse Variation. Retrieved from https://mathworld.wolfram.com/InverseVariation.html

Further Reading

• Algebra: A Comprehensive Introduction by Michael Artin
• Calculus: Early Transcendentals by James Stewart
• Mathematics for Computer Science by Eric Lehman, F Thomson Leighton, and Albert R Meyer