Suppose That $y$ Varies Inversely With $x$, And $y=6$ When $x=3$.(a) Write An Inverse Variation Equation That Relates $x$ And $y$.Equation:(b) Find $y$ When

Introduction

Inverse variation is a fundamental concept in mathematics that describes the relationship between two variables. It states that as one variable increases, the other decreases, and vice versa. In this article, we will explore the concept of inverse variation and how to write an inverse variation equation that relates two variables.

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

Example: Writing an Inverse Variation Equation

Suppose that $y$ varies inversely with $x$, and $y=6$ when $x=3$. We need to write an inverse variation equation that relates $x$ and $y$.

To do this, we can use the formula for inverse variation:

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

We are given that $y=6$ when $x=3$, so we can substitute these values into the equation:

$$6 = \frac{k}{3}$$

To solve for $k$, we can multiply both sides of the equation by 3:

$$18 = k$$

Now that we have found the value of $k$, we can write the inverse variation equation:

$$y = \frac{18}{x}$$

This equation relates $x$ and $y$ in an inverse variation relationship.

Finding $y$ When $x$ is Given

Now that we have the inverse variation equation, we can use it to find the value of $y$ when $x$ is given.

For example, suppose we want to find the value of $y$ when $x=2$. We can substitute $x=2$ into the equation:

$$y = \frac{18}{2}$$

Simplifying the equation, we get:

$$y = 9$$

Therefore, when $x=2$, the value of $y$ is 9.

Graphing Inverse Variation

Inverse variation can be graphed on a coordinate plane. The graph of an inverse variation relationship is a hyperbola, which is a curve that approaches the x-axis and y-axis but never touches them.

To graph an inverse variation relationship, we can use the equation:

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

We can choose a value of $k$ and plot the corresponding points on the coordinate plane. For example, if $k=18$, we can plot the points $(3,6)$ and $(6,3)$.

The graph of the inverse variation relationship will be a hyperbola that passes through these points.

Real-World Applications of Inverse Variation

Inverse variation has many real-world applications. For example, it can be used to model the relationship between the distance of an object from a light source and the intensity of the light.

In this scenario, the distance of the object from the light source is inversely proportional to the intensity of the light. This means that as the distance of the object from the light source increases, the intensity of the light decreases, and vice versa.

Inverse variation can also be used to model the relationship between the amount of money spent on a product and the number of units sold.

In this scenario, the amount of money spent on the product is inversely proportional to the number of units sold. This means that as the number of units sold increases, the amount of money spent on the product decreases, and vice versa.

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 decreases, and vice versa. In this article, we have explored the concept of inverse variation and how to write an inverse variation equation that relates two variables.

We have also seen how to find the value of $y$ when $x$ is given and how to graph an inverse variation relationship. Finally, we have discussed the real-world applications of inverse variation.

References

• [1] "Inverse Variation." Math Open Reference, mathopenref.com/inversevariation.html.
• [2] "Inverse Variation." Khan Academy, khanacademy.org/math/algebra/inversevariation/inversevariation.

Additional Resources

• [1] "Inverse Variation." Wolfram MathWorld, mathworld.wolfram.com/InverseVariation.html.
• [2] "Inverse Variation." Purplemath, purplemath.com/modules/inversevariation.htm.
  Inverse Variation Q&A =====================

Frequently Asked Questions About Inverse Variation

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

Q: How do I write an inverse variation equation?

A: To write an inverse variation equation, you need to know the values of $y$ and $x$ for a given point on the graph. You can use the formula:

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

to find the value of $k$, and then use that value to write the equation.

Q: How do I find the value of $y$ when $x$ is given?

A: To find the value of $y$ when $x$ is given, you can substitute the value of $x$ into the equation:

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

and solve for $y$.

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

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

Q: Can inverse variation be graphed?

A: Yes, inverse variation can be graphed on a coordinate plane. The graph of an inverse variation relationship is a hyperbola, which is a curve that approaches the x-axis and y-axis but never touches them.

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

A: Inverse variation has many real-world applications, including:

• Modeling the relationship between the distance of an object from a light source and the intensity of the light
• Modeling the relationship between the amount of money spent on a product and the number of units sold
• Modeling the relationship between the speed of an object and the time it takes to travel a certain distance

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

A: To determine if a relationship is an inverse variation, you can use the following steps:

1. Plot the points on a coordinate plane
2. Look for a hyperbola-shaped graph
3. Check if the product of the two variables is constant

Q: Can inverse variation be used to model real-world data?

A: Yes, inverse variation can be used to model real-world data. In fact, many real-world relationships can be modeled using inverse variation.

Q: What are some common mistakes to avoid when working with inverse variation?

A: Some common mistakes to avoid when working with inverse variation include:

• Confusing inverse variation with direct variation
• Not checking if the product of the two variables is constant
• Not using the correct formula to write the equation

Q: How do I choose the correct value of $k$ for an inverse variation equation?

A: To choose the correct value of $k$ for an inverse variation equation, you need to know the values of $y$ and $x$ for a given point on the graph. You can use the formula:

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

to find the value of $k$, and then use that value to write the equation.

Q: Can inverse variation be used to solve problems in physics and engineering?

A: Yes, inverse variation can be used to solve problems in physics and engineering. In fact, many real-world problems in these fields can be modeled using inverse variation.

Q: What are some other types of functional relationships besides inverse variation and direct variation?

A: Some other types of functional relationships besides inverse variation and direct variation include:

• Quadratic relationships
• Exponential relationships
• Logarithmic relationships

Q: How do I graph an inverse variation relationship on a coordinate plane?

A: To graph an inverse variation relationship on a coordinate plane, you can use the following steps:

1. Plot the points on a coordinate plane
2. Look for a hyperbola-shaped graph
3. Check if the product of the two variables is constant

Q: Can inverse variation be used to model population growth and decline?

A: Yes, inverse variation can be used to model population growth and decline. In fact, many real-world relationships between population size and other factors can be modeled using inverse variation.

Q: What are some common applications of inverse variation in economics?

A: Some common applications of inverse variation in economics include:

• Modeling the relationship between the price of a product and the quantity demanded
• Modeling the relationship between the interest rate and the amount of money borrowed
• Modeling the relationship between the inflation rate and the value of money