What Is The Constant Of Variation, { K$}$, Of The Line { Y = Kx$}$ Through The Points { (3, 18)$}$ And { (5, 30)$}$?A. { \frac{3}{5}$}$B. { \frac{5}{3}$}$C. 3D. 6
The constant of variation, denoted as {k}, is a fundamental concept in mathematics that describes the relationship between two variables in a linear equation. In this article, we will explore the concept of the constant of variation and how it can be used to find the equation of a line passing through two given points.
What is the Constant of Variation?
The constant of variation, {k$}$, is a numerical value that represents the rate of change between two variables in a linear equation. It is a measure of how much one variable changes when the other variable changes by a certain amount. In the equation {y = kx$}$, {k$}$ is the constant of variation, and it represents the slope of the line.
Finding the Constant of Variation
To find the constant of variation, we can use the formula {k = \frac{y_2 - y_1}{x_2 - x_1}$}$, where {(x_1, y_1)$}$ and {(x_2, y_2)$}$ are two points on the line. In this case, we are given the points {(3, 18)$}$ and {(5, 30)$}$.
Step 1: Identify the Points
The two points given are {(3, 18)$}$ and {(5, 30)$}$. We can use these points to find the constant of variation.
Step 2: Plug in the Values
Using the formula {k = \frac{y_2 - y_1}{x_2 - x_1}$}$, we can plug in the values of the two points:
{k = \frac{30 - 18}{5 - 3}$}$
Step 3: Simplify the Expression
Simplifying the expression, we get:
{k = \frac{12}{2}$}$
Step 4: Calculate the Value
Calculating the value, we get:
{k = 6$}$
Conclusion
Therefore, the constant of variation, {k$}$, of the line {y = kx$}$ through the points {(3, 18)$}$ and {(5, 30)$}$ is ${6\$}.
Real-World Applications
The constant of variation has many real-world applications, including:
- Physics: The constant of variation is used to describe the relationship between two physical quantities, such as distance and time.
- Economics: The constant of variation is used to describe the relationship between two economic variables, such as price and quantity demanded.
- Engineering: The constant of variation is used to describe the relationship between two engineering variables, such as voltage and current.
Common Mistakes
When finding the constant of variation, there are several common mistakes to avoid:
- Incorrectly identifying the points: Make sure to identify the correct points on the line.
- Incorrectly plugging in the values: Make sure to plug in the correct values into the formula.
- Incorrectly simplifying the expression: Make sure to simplify the expression correctly.
Conclusion
Q: What is the constant of variation?
A: The constant of variation, denoted as {k$}$, is a numerical value that represents the rate of change between two variables in a linear equation. It is a measure of how much one variable changes when the other variable changes by a certain amount.
Q: How do I find the constant of variation?
A: To find the constant of variation, you can use the formula {k = \frac{y_2 - y_1}{x_2 - x_1}$}$, where {(x_1, y_1)$}$ and {(x_2, y_2)$}$ are two points on the line.
Q: What are the two points needed to find the constant of variation?
A: The two points needed to find the constant of variation are any two points on the line. For example, if you are given the points {(3, 18)$}$ and {(5, 30)$}$, you can use these points to find the constant of variation.
Q: How do I plug in the values into the formula?
A: To plug in the values into the formula, simply substitute the values of the two points into the formula. For example, if you are given the points {(3, 18)$}$ and {(5, 30)$}$, you would plug in the values as follows:
{k = \frac{30 - 18}{5 - 3}$}$
Q: What if I get a negative value for the constant of variation?
A: If you get a negative value for the constant of variation, it means that the line is decreasing as the x-value increases. This is a valid result, and it simply means that the line is sloping downward.
Q: Can I use the constant of variation to find the equation of a line?
A: Yes, you can use the constant of variation to find the equation of a line. Once you have found the constant of variation, you can use it to write the equation of the line in the form {y = kx$}$.
Q: What are some real-world applications of the constant of variation?
A: The constant of variation has many real-world applications, including:
- Physics: The constant of variation is used to describe the relationship between two physical quantities, such as distance and time.
- Economics: The constant of variation is used to describe the relationship between two economic variables, such as price and quantity demanded.
- Engineering: The constant of variation is used to describe the relationship between two engineering variables, such as voltage and current.
Q: What are some common mistakes to avoid when finding the constant of variation?
A: Some common mistakes to avoid when finding the constant of variation include:
- Incorrectly identifying the points: Make sure to identify the correct points on the line.
- Incorrectly plugging in the values: Make sure to plug in the correct values into the formula.
- Incorrectly simplifying the expression: Make sure to simplify the expression correctly.
Conclusion
In conclusion, the constant of variation is a fundamental concept in mathematics that describes the relationship between two variables in a linear equation. By using the formula {k = \frac{y_2 - y_1}{x_2 - x_1}$}$, we can find the constant of variation of a line passing through two given points. The constant of variation has many real-world applications, and it is an important concept to understand in mathematics.