Which Equation Represents An Inverse Variation?A. Y = 2 X Y = 2x Y = 2 X B. Y = X 3 Y = \frac{x}{3} Y = 3 X ​ C. Y = 4 X Y = \frac{4}{x} Y = X 4 ​ D. Y = − 5 X Y = -5x Y = − 5 X

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Inverse variation is a fundamental concept in mathematics that describes the relationship between two variables. It is a type of variation where the product of the two variables remains constant. In this article, we will explore the concept of inverse variation and identify which equation represents an inverse variation.

What is Inverse Variation?

Inverse variation is a relationship between two variables, x and y, where the product of the two variables remains constant. This means that as one variable increases, the other variable decreases, and vice versa. The equation that represents inverse variation is:

y = k/x

where k is a constant.

Characteristics of Inverse Variation

Inverse variation has several characteristics that distinguish it from other types of variation. Some of the key characteristics of inverse variation include:

  • The product of the two variables remains constant.
  • As one variable increases, the other variable decreases.
  • The relationship is non-linear.
  • The graph of the equation is a hyperbola.

Examples of Inverse Variation

Inverse variation can be seen in many real-world situations. Some examples include:

  • The relationship between the distance of an object from a light source and the intensity of the light it receives.
  • The relationship between the amount of money spent on a product and the number of units sold.
  • The relationship between the time it takes to complete a task and the number of workers assigned to it.

Which Equation Represents an Inverse Variation?

Now that we have a good understanding of inverse variation, let's examine the options provided:

A. y=2xy = 2x B. y=x3y = \frac{x}{3} C. y=4xy = \frac{4}{x} D. y=−5xy = -5x

To determine which equation represents an inverse variation, we need to examine each option and identify the characteristics of inverse variation.

  • Option A: y=2xy = 2x is a linear equation, not an inverse variation.
  • Option B: y=x3y = \frac{x}{3} is a linear equation, not an inverse variation.
  • Option C: y=4xy = \frac{4}{x} is an inverse variation equation, as it meets the characteristics of inverse variation.
  • Option D: y=−5xy = -5x is a linear equation, not an inverse variation.

Conclusion

In conclusion, the equation that represents an inverse variation is:

y = k/x

where k is a constant. This equation meets the characteristics of inverse variation, including the product of the two variables remaining constant, the relationship being non-linear, and the graph being a hyperbola. The correct answer is:

C. y=4xy = \frac{4}{x}

This equation represents an inverse variation, as it meets the characteristics of inverse variation.

Additional Examples and Practice Problems

Here are some additional examples and practice problems to help you understand inverse variation better:

  • Example 1: A company produces x units of a product and sells them for $y dollars each. If the total revenue is $120, find the value of x.
  • Example 2: A car travels at a speed of x miles per hour and covers a distance of y miles. If the time taken is 2 hours, find the value of x.
  • Practice Problem 1: A person invests $x dollars in a savings account that earns an interest rate of 5% per year. If the total amount after 2 years is $120, find the value of x.
  • Practice Problem 2: A company produces x units of a product and sells them for $y dollars each. If the total revenue is $150, find the value of x.

Solutions to Additional Examples and Practice Problems

Here are the solutions to the additional examples and practice problems:

  • Example 1: A company produces x units of a product and sells them for $y dollars each. If the total revenue is $120, find the value of x.
    • Solution: Let's use the equation y = k/x, where k is a constant. We know that the total revenue is $120, so we can set up the equation: 120 = k/x
    • To solve for x, we can multiply both sides by x: 120x = k
    • Since k is a constant, we can divide both sides by 120: x = k/120
    • Therefore, the value of x is k/120.
  • Example 2: A car travels at a speed of x miles per hour and covers a distance of y miles. If the time taken is 2 hours, find the value of x.
    • Solution: Let's use the equation y = k/x, where k is a constant. We know that the time taken is 2 hours, so we can set up the equation: 2 = k/x
    • To solve for x, we can multiply both sides by x: 2x = k
    • Since k is a constant, we can divide both sides by 2: x = k/2
    • Therefore, the value of x is k/2.
  • Practice Problem 1: A person invests $x dollars in a savings account that earns an interest rate of 5% per year. If the total amount after 2 years is $120, find the value of x.
    • Solution: Let's use the equation y = k/x, where k is a constant. We know that the total amount after 2 years is $120, so we can set up the equation: 120 = k/x
    • To solve for x, we can multiply both sides by x: 120x = k
    • Since k is a constant, we can divide both sides by 120: x = k/120
    • Therefore, the value of x is k/120.
  • Practice Problem 2: A company produces x units of a product and sells them for $y dollars each. If the total revenue is $150, find the value of x.
    • Solution: Let's use the equation y = k/x, where k is a constant. We know that the total revenue is $150, so we can set up the equation: 150 = k/x
    • To solve for x, we can multiply both sides by x: 150x = k
    • Since k is a constant, we can divide both sides by 150: x = k/150
    • Therefore, the value of x is k/150.

Conclusion

In conclusion, the equation that represents an inverse variation is:

y = k/x

where k is a constant. This equation meets the characteristics of inverse variation, including the product of the two variables remaining constant, the relationship being non-linear, and the graph being a hyperbola. The correct answer is:

C. y=4xy = \frac{4}{x}

Inverse variation is a fundamental concept in mathematics that describes the relationship between two variables. In this article, we will answer some frequently asked questions about inverse variation.

Q: What is inverse variation?

A: Inverse variation is a type of variation where the product of the two variables remains constant. This means that as one variable increases, the other variable decreases, and vice versa.

Q: What are the characteristics of inverse variation?

A: The characteristics of inverse variation include:

  • The product of the two variables remains constant.
  • As one variable increases, the other variable decreases.
  • The relationship is non-linear.
  • The graph of the equation is a hyperbola.

Q: What are some examples of inverse variation?

A: Some examples of inverse variation include:

  • The relationship between the distance of an object from a light source and the intensity of the light it receives.
  • The relationship between the amount of money spent on a product and the number of units sold.
  • The relationship between the time it takes to complete a task and the number of workers assigned to it.

Q: How do I identify an inverse variation equation?

A: To identify an inverse variation equation, look for the following characteristics:

  • The equation is in the form y = k/x, where k is a constant.
  • The product of the two variables remains constant.
  • The relationship is non-linear.
  • The graph of the equation is a hyperbola.

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

A: Inverse variation and direct variation are two types of variation that describe the relationship between two variables. In direct variation, the product of the two variables increases as one variable increases. In inverse variation, the product of the two variables remains constant.

Q: Can you provide some examples of inverse variation equations?

A: Here are some examples of inverse variation equations:

  • y = 4/x
  • y = 2/x
  • y = 3/x
  • y = 5/x

Q: How do I solve an inverse variation equation?

A: To solve an inverse variation equation, follow these steps:

  1. Identify the equation and the variables.
  2. Multiply both sides of the equation by the variable to eliminate the fraction.
  3. Simplify the equation and solve for the variable.

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

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

  • Physics: The relationship between the distance of an object from a light source and the intensity of the light it receives.
  • Economics: The relationship between the amount of money spent on a product and the number of units sold.
  • Engineering: The relationship between the time it takes to complete a task and the number of workers assigned to it.

Q: Can you provide some practice problems for inverse variation?

A: Here are some practice problems for inverse variation:

  • Problem 1: A company produces x units of a product and sells them for $y dollars each. If the total revenue is $120, find the value of x.
  • Problem 2: A car travels at a speed of x miles per hour and covers a distance of y miles. If the time taken is 2 hours, find the value of x.
  • Problem 3: A person invests $x dollars in a savings account that earns an interest rate of 5% per year. If the total amount after 2 years is $120, find the value of x.

Solutions to Practice Problems

Here are the solutions to the practice problems:

  • Problem 1: A company produces x units of a product and sells them for $y dollars each. If the total revenue is $120, find the value of x.
    • Solution: Let's use the equation y = k/x, where k is a constant. We know that the total revenue is $120, so we can set up the equation: 120 = k/x
    • To solve for x, we can multiply both sides by x: 120x = k
    • Since k is a constant, we can divide both sides by 120: x = k/120
    • Therefore, the value of x is k/120.
  • Problem 2: A car travels at a speed of x miles per hour and covers a distance of y miles. If the time taken is 2 hours, find the value of x.
    • Solution: Let's use the equation y = k/x, where k is a constant. We know that the time taken is 2 hours, so we can set up the equation: 2 = k/x
    • To solve for x, we can multiply both sides by x: 2x = k
    • Since k is a constant, we can divide both sides by 2: x = k/2
    • Therefore, the value of x is k/2.
  • Problem 3: A person invests $x dollars in a savings account that earns an interest rate of 5% per year. If the total amount after 2 years is $120, find the value of x.
    • Solution: Let's use the equation y = k/x, where k is a constant. We know that the total amount after 2 years is $120, so we can set up the equation: 120 = k/x
    • To solve for x, we can multiply both sides by x: 120x = k
    • Since k is a constant, we can divide both sides by 120: x = k/120
    • Therefore, the value of x is k/120.

Conclusion

In conclusion, inverse variation is a fundamental concept in mathematics that describes the relationship between two variables. It has many real-world applications, including physics, economics, and engineering. To identify an inverse variation equation, look for the characteristics of inverse variation, including the product of the two variables remaining constant, the relationship being non-linear, and the graph of the equation being a hyperbola. To solve an inverse variation equation, follow the steps outlined above.