The Variables { X $}$ And { Y $}$ Vary Directly. Use The Given Values To Write An Equation That Relates { X $}$ And { Y $} . . . {$ X = 18 $}$ { Y = 6 $}$ A. [$ Y = \frac{2}{3} X
What is Direct Variation?
Direct variation is a relationship between two variables, x and y, where the value of y is directly proportional to the value of x. This means that as the value of x increases, the value of y also increases, and vice versa. The relationship between x and y can be represented by the equation y = kx, where k is the constant of proportionality.
Given Values
We are given two values: x = 18 and y = 6. These values will help us write an equation that relates x and y.
Writing the Equation
To write the equation, we need to find the constant of proportionality, k. We can do this by substituting the given values into the equation y = kx.
y = kx 6 = k(18)
Solving for k
To solve for k, we need to isolate k on one side of the equation. We can do this by dividing both sides of the equation by 18.
k = 6/18 k = 1/3
The Equation of Direct Variation
Now that we have found the constant of proportionality, k, we can write the equation of direct variation.
y = kx y = (1/3)x
Simplifying the Equation
We can simplify the equation by multiplying both sides by 3 to get rid of the fraction.
y = (1/3)x 3y = x
The Final Equation
The final equation that relates x and y is:
y = (1/3)x
Alternative Equation
We can also write the equation in the form y = mx, where m is the slope of the line.
y = (1/3)x y = (2/3)x
Conclusion
In this article, we have learned how to write an equation that relates two variables, x and y, when they vary directly. We have used the given values x = 18 and y = 6 to find the constant of proportionality, k, and have written the equation of direct variation in the form y = kx. We have also simplified the equation and written it in the form y = mx.
Example Problems
- If x = 12 and y = 4, write an equation that relates x and y.
- If x = 24 and y = 8, write an equation that relates x and y.
- If x = 36 and y = 12, write an equation that relates x and y.
Answer Key
- y = (1/3)x
- y = (1/3)x
- y = (1/3)x
Tips and Tricks
- When writing an equation of direct variation, make sure to find the constant of proportionality, k.
- Use the given values to substitute into the equation and solve for k.
- Simplify the equation by multiplying both sides by the denominator to get rid of the fraction.
- Write the equation in the form y = mx, where m is the slope of the line.
Common Mistakes
- Failing to find the constant of proportionality, k.
- Not using the given values to substitute into the equation.
- Not simplifying the equation by multiplying both sides by the denominator.
- Writing the equation in the wrong form, such as y = kx instead of y = mx.
The Variables x and y Vary Directly: Q&A =====================================================
Q: What is direct variation?
A: Direct variation is a relationship between two variables, x and y, where the value of y is directly proportional to the value of x. This means that as the value of x increases, the value of y also increases, and vice versa.
Q: How do I write an equation of direct variation?
A: To write an equation of direct variation, you need to find the constant of proportionality, k. You can do this by substituting the given values into the equation y = kx. Then, solve for k and simplify the equation.
Q: What is the constant of proportionality, k?
A: The constant of proportionality, k, is a number that represents the ratio of y to x. It is a measure of how much y changes when x changes.
Q: How do I find the constant of proportionality, k?
A: To find the constant of proportionality, k, you need to substitute the given values into the equation y = kx and solve for k. You can do this by dividing both sides of the equation by x.
Q: What is the equation of direct variation in the form y = mx?
A: The equation of direct variation in the form y = mx is a simplified version of the equation y = kx. It is written in the form y = mx, where m is the slope of the line.
Q: How do I simplify the equation y = kx?
A: To simplify the equation y = kx, you need to multiply both sides of the equation by the denominator to get rid of the fraction.
Q: What are some common mistakes to avoid when writing an equation of direct variation?
A: Some common mistakes to avoid when writing an equation of direct variation include:
- Failing to find the constant of proportionality, k.
- Not using the given values to substitute into the equation.
- Not simplifying the equation by multiplying both sides by the denominator.
- Writing the equation in the wrong form, such as y = kx instead of y = mx.
Q: How do I use the equation of direct variation to solve problems?
A: To use the equation of direct variation to solve problems, you need to substitute the given values into the equation and solve for the unknown variable.
Q: What are some real-world applications of direct variation?
A: Some real-world applications of direct variation include:
- Modeling the relationship between the price of a product and the quantity demanded.
- Modeling the relationship between the speed of an object and the distance traveled.
- Modeling the relationship between the amount of money invested and the return on investment.
Q: How do I determine if two variables are in direct variation?
A: To determine if two variables are in direct variation, you need to check if the ratio of y to x is constant. If the ratio is constant, then the variables are in direct variation.
Q: What is the difference between direct variation and inverse variation?
A: The difference between direct variation and inverse variation is that in direct variation, the value of y increases as the value of x increases, while in inverse variation, the value of y decreases as the value of x increases.
Q: How do I write an equation of inverse variation?
A: To write an equation of inverse variation, you need to find the constant of proportionality, k, and then write the equation in the form y = k/x.
Q: What are some common mistakes to avoid when writing an equation of inverse variation?
A: Some common mistakes to avoid when writing an equation of inverse variation include:
- Failing to find the constant of proportionality, k.
- Not using the given values to substitute into the equation.
- Not simplifying the equation by multiplying both sides by the denominator.
- Writing the equation in the wrong form, such as y = kx instead of y = k/x.