Using the Given Values to Write an Equation

Oct 13, 2025 by recoveryshake.com 124 views

The Variables ${$ x$}$ and ${$ y$}$ Vary Directly: Understanding the Relationship

When two variables vary directly, it means that as one variable increases, the other variable also increases at a constant rate. This type of relationship is often represented by a linear equation, where the variables are directly proportional to each other. In this article, we will explore the concept of direct variation and use the given values to write an equation that relates the variables ${$ x$}$ and ${$ y$}$.

What is Direct Variation?

Direct variation is a type of relationship between two variables where the ratio of the variables remains constant. This means that if we multiply one variable by a constant, the other variable will also be multiplied by the same constant. Mathematically, this can be represented as:

y = kx

where ${$ y$}$ is the dependent variable, ${$ x$}$ is the independent variable, and ${$ k$}$ is the constant of proportionality.

Using the Given Values to Write an Equation

We are given the values ${$ x = 20$}$ and ${$ y = 10$}$. To write an equation that relates these variables, we need to find the constant of proportionality ${$ k$}$. We can do this by substituting the given values into the equation:

10 = k(20)

To solve for ${$ k$}$, we can divide both sides of the equation by 20:

k = \frac{10}{20}

k = \frac{1}{2}

Now that we have found the constant of proportionality, we can write the equation that relates the variables ${$ x$}$ and ${$ y$}$:

y = \frac{1}{2}x

Simplifying the Equation

We can simplify the equation by multiplying both sides by 2:

2y = x

This is the simplified equation that relates the variables ${$ x$}$ and ${$ y$}$.

Comparing with the Given Options

Let's compare our simplified equation with the given options:

A. ${$ y = x^2$}$

B. ${$ y = \frac{1}{2}x$}$

Our simplified equation matches option B, which is ${$ y = \frac{1}{2}x$}$.

Conclusion

In this article, we explored the concept of direct variation and used the given values to write an equation that relates the variables ${$ x$}$ and ${$ y$}$. We found that the constant of proportionality is ${$ \frac{1}{2}$}$ and wrote the equation ${$ y = \frac{1}{2}x$}$. This equation represents the direct variation relationship between the variables ${$ x$}$ and ${$ y$}$.

Understanding the Relationship

The equation ${$ y = \frac{1}{2}x$}$ represents a direct variation relationship between the variables ${$ x$}$ and ${$ y$}$. This means that as ${$ x$}$ increases, ${$ y$}$ also increases at a constant rate. For example, if ${$ x$}$ is doubled, ${$ y$}$ will also be doubled.

Real-World Applications

Direct variation relationships are common in real-world applications, such as:

The cost of goods sold: The cost of goods sold may vary directly with the quantity sold.
The speed of an object: The speed of an object may vary directly with the force applied.
The volume of a gas: The volume of a gas may vary directly with the temperature.

Conclusion

In conclusion, the variables ${$ x$}$ and ${$ y$}$ vary directly, and we have written an equation that relates these variables. The equation ${$ y = \frac{1}{2}x$}$ represents the direct variation relationship between the variables ${$ x$}$ and ${$ y$}$. This equation can be used to solve problems involving direct variation relationships.

Frequently Asked Questions

Q: What is direct variation? A: Direct variation is a type of relationship between two variables where the ratio of the variables remains constant.
Q: How do I write an equation that relates two variables that vary directly? A: To write an equation that relates two variables that vary directly, you need to find the constant of proportionality and use it to write the equation.
Q: What is the constant of proportionality? A: The constant of proportionality is a value that represents the ratio of the variables in a direct variation relationship.

References

[1] "Direct Variation." Math Open Reference, mathopenref.com/directvariation.html.
[2] "Direct Variation." Khan Academy, khanacademy.org/math/algebra/x2f6f6c6c/x2f6f6c6c-article.
[3] "Direct Variation." Wolfram MathWorld, mathworld.wolfram.com/DirectVariation.html.
Direct Variation Q&A: Frequently Asked Questions

In our previous article, we explored the concept of direct variation and used the given values to write an equation that relates the variables ${$ x$}$ and ${$ y$}$. In this article, we will answer some frequently asked questions about direct variation.

Q: What is direct variation?

A: Direct variation is a type of relationship between two variables where the ratio of the variables remains constant. This means that if we multiply one variable by a constant, the other variable will also be multiplied by the same constant.

Q: How do I know if two variables vary directly?

A: To determine if two variables vary directly, you need to check if the ratio of the variables remains constant. You can do this by plotting the variables on a graph and checking if the points lie on a straight line.

Q: What is the equation of a direct variation relationship?

A: The equation of a direct variation relationship is given by:

y = kx

where ${$ y$}$ is the dependent variable, ${$ x$}$ is the independent variable, and ${$ k$}$ is the constant of proportionality.

Q: How do I find the constant of proportionality?

A: To find the constant of proportionality, you need to substitute the given values into the equation and solve for ${$ k$}$. For example, if we are given the values ${$ x = 20$}$ and ${$ y = 10$}$, we can substitute these values into the equation:

10 = k(20)

To solve for ${$ k$}$, we can divide both sides of the equation by 20:

k = \frac{10}{20}

k = \frac{1}{2}

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

A: Direct variation and inverse variation are two types of relationships between variables. In direct variation, the ratio of the variables remains constant, while in inverse variation, the product of the variables remains constant.

Q: Can you give an example of direct variation in real life?

A: Yes, direct variation is common in real life. For example, the cost of goods sold may vary directly with the quantity sold. If the quantity sold increases, the cost of goods sold will also increase at a constant rate.

Q: How do I graph a direct variation relationship?

A: To graph a direct variation relationship, you need to plot the variables on a graph and check if the points lie on a straight line. If the points lie on a straight line, then the relationship is a direct variation.

Q: Can you give an example of a direct variation equation?

A: Yes, here is an example of a direct variation equation:

y = 2x

This equation represents a direct variation relationship between the variables ${$ x$}$ and ${$ y$}$.

Q: What is the significance of direct variation in science and engineering?

A: Direct variation is significant in science and engineering because it helps us understand the relationships between variables in different systems. For example, in physics, direct variation is used to describe the relationship between force and acceleration.

Q: Can you give an example of a direct variation problem?

A: Yes, here is an example of a direct variation problem:

A car travels 250 miles in 5 hours. How far will it travel in 10 hours?

To solve this problem, we need to find the constant of proportionality and use it to write an equation that relates the variables. Let's say the distance traveled is ${$ y$}$ and the time taken is ${$ x$}$. We can write the equation:

y = kx

Substituting the given values, we get:

250 = k(5)

To solve for ${$ k$}$, we can divide both sides of the equation by 5:

k = \frac{250}{5}

k = 50

Now that we have found the constant of proportionality, we can write the equation:

y = 50x

Substituting ${$ x = 10$}$, we get:

y = 50(10)

y = 500

Therefore, the car will travel 500 miles in 10 hours.

Conclusion

In this article, we have answered some frequently asked questions about direct variation. We have discussed the concept of direct variation, how to determine if two variables vary directly, and how to find the constant of proportionality. We have also given examples of direct variation equations and problems.